Custom Cell Reorder Behavior in CollectionView

I managed to achieve this by creating a subclass of UICollectionView and adding custom handling to interactive movement. While looking at possible hints on how to solve your issue, I've found this tutorial : http://nshint.io/blog/2015/07/16/uicollectionviews-now-have-easy-reordering/. The most important part there was that interactive reordering can be done not only on UICollectionViewController. The relevant code looks like this :

var longPressGesture : UILongPressGestureRecognizer!

override func viewDidLoad() {
    super.viewDidLoad()

    // rest of setup        

    longPressGesture = UILongPressGestureRecognizer(target: self, action: #selector(ViewController.handleLongGesture(_:)))
    self.collectionView?.addGestureRecognizer(longPressGesture)

}

func handleLongGesture(gesture: UILongPressGestureRecognizer) {

    switch(gesture.state) {

    case UIGestureRecognizerState.Began:
        guard let selectedIndexPath = self.collectionView?.indexPathForItemAtPoint(gesture.locationInView(self.collectionView)) else {
            break
        }
        collectionView?.beginInteractiveMovementForItemAtIndexPath(selectedIndexPath)
    case UIGestureRecognizerState.Changed:
        collectionView?.updateInteractiveMovementTargetPosition(gesture.locationInView(gesture.view!))
    case UIGestureRecognizerState.Ended:
        collectionView?.endInteractiveMovement()
    default:
        collectionView?.cancelInteractiveMovement()
    }
}

This needs to be inside your view controller in which your collection view is placed. I don't know if this will work with UICollectionViewController, some additional tinkering may be needed. What led me to subclassing UICollectionView was realisation that all other related classes/delegate methods are informed only about the first and last index paths (i.e. the source and destination), and there is no information about all the other cells that got rearranged, so It needed to be stopped at the core.

SwappingCollectionView.swift :

import UIKit

extension UIView {
    func snapshot() -> UIImage {
        UIGraphicsBeginImageContext(self.bounds.size)
        self.layer.renderInContext(UIGraphicsGetCurrentContext()!)
        let image = UIGraphicsGetImageFromCurrentImageContext()
        UIGraphicsEndImageContext()
        return image
    }
}

extension CGPoint {
    func distanceToPoint(p:CGPoint) -> CGFloat {
        return sqrt(pow((p.x - x), 2) + pow((p.y - y), 2))
    }
}

struct SwapDescription : Hashable {
    var firstItem : Int
    var secondItem : Int

    var hashValue: Int {
        get {
            return (firstItem * 10) + secondItem
        }
    }
}

func ==(lhs: SwapDescription, rhs: SwapDescription) -> Bool {
    return lhs.firstItem == rhs.firstItem && lhs.secondItem == rhs.secondItem
}

class SwappingCollectionView: UICollectionView {

    var interactiveIndexPath : NSIndexPath?
    var interactiveView : UIView?
    var interactiveCell : UICollectionViewCell?
    var swapSet : Set<SwapDescription> = Set()
    var previousPoint : CGPoint?

    static let distanceDelta:CGFloat = 2 // adjust as needed

    override func beginInteractiveMovementForItemAtIndexPath(indexPath: NSIndexPath) -> Bool {

        self.interactiveIndexPath = indexPath

        self.interactiveCell = self.cellForItemAtIndexPath(indexPath)

        self.interactiveView = UIImageView(image: self.interactiveCell?.snapshot())
        self.interactiveView?.frame = self.interactiveCell!.frame

        self.addSubview(self.interactiveView!)
        self.bringSubviewToFront(self.interactiveView!)

        self.interactiveCell?.hidden = true

        return true
    }

    override func updateInteractiveMovementTargetPosition(targetPosition: CGPoint) {

        if (self.shouldSwap(targetPosition)) {

            if let hoverIndexPath = self.indexPathForItemAtPoint(targetPosition), let interactiveIndexPath = self.interactiveIndexPath {

                let swapDescription = SwapDescription(firstItem: interactiveIndexPath.item, secondItem: hoverIndexPath.item)

                if (!self.swapSet.contains(swapDescription)) {

                    self.swapSet.insert(swapDescription)

                    self.performBatchUpdates({
                        self.moveItemAtIndexPath(interactiveIndexPath, toIndexPath: hoverIndexPath)
                        self.moveItemAtIndexPath(hoverIndexPath, toIndexPath: interactiveIndexPath)
                        }, completion: {(finished) in
                            self.swapSet.remove(swapDescription)                                
                            self.dataSource?.collectionView(self, moveItemAtIndexPath: interactiveIndexPath, toIndexPath: hoverIndexPath)
                            self.interactiveIndexPath = hoverIndexPath

                    })
                }
            }
        }

        self.interactiveView?.center = targetPosition
        self.previousPoint = targetPosition
    }

    override func endInteractiveMovement() {
        self.cleanup()
    }

    override func cancelInteractiveMovement() {
        self.cleanup()
    }

    func cleanup() {
        self.interactiveCell?.hidden = false
        self.interactiveView?.removeFromSuperview()
        self.interactiveView = nil
        self.interactiveCell = nil
        self.interactiveIndexPath = nil
        self.previousPoint = nil
        self.swapSet.removeAll()
    }

    func shouldSwap(newPoint: CGPoint) -> Bool {
        if let previousPoint = self.previousPoint {
            let distance = previousPoint.distanceToPoint(newPoint)
            return distance < SwappingCollectionView.distanceDelta
        }

        return false
    }
}

I do realize that there is a lot going on there, but I hope everything will be clear in a minute.

1. Extension on UIView with helper method to get a snapshot of a cell.
2. Extension on CGPoint with helper method to calculate distance between two points.
3. SwapDescription helper structure - it is needed to prevent multiple swaps of the same pair of items, which resulted in glitchy animations. Its hashValue method could be improved, but was good enough for the sake of this proof of concept.
4. beginInteractiveMovementForItemAtIndexPath(indexPath: NSIndexPath) -> Bool - the method called when the movement begins. Everything gets setup here. We get a snapshot of our cell and add it as a subview - this snapshot will be what the user actually drags on screen. The cell itself gets hidden. If you return false from this method, the interactive movement will not begin.
5. updateInteractiveMovementTargetPosition(targetPosition: CGPoint) - method called after each user movement, which is a lot. We check if the distance from previous point is small enough to swap items - this prevents issue when the user would drag fast across screen and multiple items would get swapped with non-obvious results. If the swap can happen, we check if it is already happening, and if not we swap two items.
6. endInteractiveMovement(), cancelInteractiveMovement(), cleanup() - after the movement ends, we need to restore our helpers to their default state.
7. shouldSwap(newPoint: CGPoint) -> Bool - helper method to check if the movement was small enough so we can swap cells.

This is the result it gives :

Let me know if this is what you needed and/or if you need me to clarify something.

Here is a demo project.

Swift 5 solution of @Losiowaty solution:

var longPressGesture : UILongPressGestureRecognizer!

override func viewDidLoad()
{
    super.viewDidLoad()

    // rest of setup
    longPressGesture = UILongPressGestureRecognizer(target: self, action: #selector(handleLongGesture))
    self.collectionView?.addGestureRecognizer(longPressGesture)
}

@objc func handleLongGesture(gesture: UILongPressGestureRecognizer)
{
    switch(gesture.state)
    {
    case UIGestureRecognizerState.began:
        guard let selectedIndexPath = self.collectionView?.indexPathForItem(at: gesture.location(in: self.collectionView)) else {
            break
        }
        collectionView?.beginInteractiveMovementForItem(at: selectedIndexPath)
    case UIGestureRecognizerState.changed:
        collectionView?.updateInteractiveMovementTargetPosition(gesture.location(in: gesture.view!))
    case UIGestureRecognizerState.ended:
        collectionView?.endInteractiveMovement()
    default:
        collectionView?.cancelInteractiveMovement()
    }
}


import UIKit

extension UIView {
    func snapshot() -> UIImage {
        UIGraphicsBeginImageContext(self.bounds.size)
        self.layer.render(in: UIGraphicsGetCurrentContext()!)
        let image = UIGraphicsGetImageFromCurrentImageContext()
        UIGraphicsEndImageContext()
        return image!
    }
}

extension CGPoint {
    func distanceToPoint(p:CGPoint) -> CGFloat {
        return sqrt(pow((p.x - x), 2) + pow((p.y - y), 2))
    }
}

struct SwapDescription : Hashable {
    var firstItem : Int
    var secondItem : Int

    var hashValue: Int {
        get {
            return (firstItem * 10) + secondItem
        }
    }
}

func ==(lhs: SwapDescription, rhs: SwapDescription) -> Bool {
    return lhs.firstItem == rhs.firstItem && lhs.secondItem == rhs.secondItem
}

class SwappingCollectionView: UICollectionView {

    var interactiveIndexPath : IndexPath?
    var interactiveView : UIView?
    var interactiveCell : UICollectionViewCell?
    var swapSet : Set<SwapDescription> = Set()
    var previousPoint : CGPoint?

    static let distanceDelta:CGFloat = 2 // adjust as needed

    override func beginInteractiveMovementForItem(at indexPath: IndexPath) -> Bool
    {
        self.interactiveIndexPath = indexPath

        self.interactiveCell = self.cellForItem(at: indexPath)

        self.interactiveView = UIImageView(image: self.interactiveCell?.snapshot())
        self.interactiveView?.frame = self.interactiveCell!.frame

        self.addSubview(self.interactiveView!)
        self.bringSubviewToFront(self.interactiveView!)

        self.interactiveCell?.isHidden = true

        return true
    }

    override func updateInteractiveMovementTargetPosition(_ targetPosition: CGPoint) {

        if (self.shouldSwap(newPoint: targetPosition))
        {
            if let hoverIndexPath = self.indexPathForItem(at: targetPosition), let interactiveIndexPath = self.interactiveIndexPath {

                let swapDescription = SwapDescription(firstItem: interactiveIndexPath.item, secondItem: hoverIndexPath.item)

                if (!self.swapSet.contains(swapDescription)) {

                    self.swapSet.insert(swapDescription)

                    self.performBatchUpdates({
                        self.moveItem(at: interactiveIndexPath as IndexPath, to: hoverIndexPath)
                        self.moveItem(at: hoverIndexPath, to: interactiveIndexPath)
                        }, completion: {(finished) in
                            self.swapSet.remove(swapDescription)
                            self.dataSource?.collectionView?(self, moveItemAt: interactiveIndexPath, to: hoverIndexPath)
                            self.interactiveIndexPath = hoverIndexPath

                    })
                }
            }
        }

        self.interactiveView?.center = targetPosition
        self.previousPoint = targetPosition
    }

    override func endInteractiveMovement() {
        self.cleanup()
    }

    override func cancelInteractiveMovement() {
        self.cleanup()
    }

    func cleanup() {
        self.interactiveCell?.isHidden = false
        self.interactiveView?.removeFromSuperview()
        self.interactiveView = nil
        self.interactiveCell = nil
        self.interactiveIndexPath = nil
        self.previousPoint = nil
        self.swapSet.removeAll()
    }

    func shouldSwap(newPoint: CGPoint) -> Bool {
        if let previousPoint = self.previousPoint {
            let distance = previousPoint.distanceToPoint(p: newPoint)
            return distance < SwappingCollectionView.distanceDelta
        }

        return false
    }
}