How to: Using Combine to react to CoreData changes in the background

You can pass the object you want to observe to publisher(for:):

NotificationCenter.default
  .publisher(for: .NSManagedObjectContextDidSave, object: backgroundMoc)
  .sink(receiveValue: { notification in
    // handle changes
  })

That will only listen for notifications related to a background managed object context which means you can do processing on that context's queue safely.

You can create a Publisher which informs you when something relevant for you in Core Data has changed.

I wrote an article on this. Combine, Publishers and Core Data.

import Combine
import CoreData
import Foundation

class CDPublisher<Entity>: NSObject, NSFetchedResultsControllerDelegate, Publisher where Entity: NSManagedObject {
    typealias Output = [Entity]
    typealias Failure = Error

    private let request: NSFetchRequest<Entity>
    private let context: NSManagedObjectContext
    private let subject: CurrentValueSubject<[Entity], Failure>
    private var resultController: NSFetchedResultsController<NSManagedObject>?
    private var subscriptions = 0

      init(request: NSFetchRequest<Entity>, context: NSManagedObjectContext) {
        if request.sortDescriptors == nil { request.sortDescriptors = [] }
        self.request = request
        self.context = context
        subject = CurrentValueSubject([])
        super.init()
    }

      func receive<S>(subscriber: S)
        where S: Subscriber, CDPublisher.Failure == S.Failure, CDPublisher.Output == S.Input {
        var start = false

        synchronized(self) {
            subscriptions += 1
            start = subscriptions == 1
        }

        if start {
            let controller = NSFetchedResultsController(fetchRequest: request, managedObjectContext: context, 
                                                        sectionNameKeyPath: nil, cacheName: nil)
            controller.delegate = self

            do {
                try controller.performFetch()
                let result = controller.fetchedObjects ?? []
                subject.send(result)
            } catch {
                subject.send(completion: .failure(error))
            }
            resultController = controller as? NSFetchedResultsController<NSManagedObject>
        }
        CDSubscription(fetchPublisher: self, subscriber: AnySubscriber(subscriber))
    }

      func controllerDidChangeContent(_ controller: NSFetchedResultsController<NSFetchRequestResult>) {
        let result = controller.fetchedObjects as? [Entity] ?? []
        subject.send(result)
    }

      private func dropSubscription() {
        objc_sync_enter(self)
        subscriptions -= 1
        let stop = subscriptions == 0
        objc_sync_exit(self)

        if stop {
            resultController?.delegate = nil
            resultController = nil
        }
    }

    private class CDSubscription: Subscription {
        private var fetchPublisher: CDPublisher?
        private var cancellable: AnyCancellable?

        @discardableResult
        init(fetchPublisher: CDPublisher, subscriber: AnySubscriber<Output, Failure>) {
            self.fetchPublisher = fetchPublisher

            subscriber.receive(subscription: self)

            cancellable = fetchPublisher.subject.sink(receiveCompletion: { completion in
                subscriber.receive(completion: completion)
            }, receiveValue: { value in
                _ = subscriber.receive(value)
            })
        }

        func request(_ demand: Subscribers.Demand) {}

        func cancel() {
            cancellable?.cancel()
            cancellable = nil
            fetchPublisher?.dropSubscription()
            fetchPublisher = nil
        }
    }

}