lodash property search in array and in nested child arrays

That's a very simple tree traversal task. The easiest way to solve it is recursion (link to jsbin). It will work with any depth (with recursion limit of course) and it's one of the fastest ways with the worst complexity O(n):

function find(id, items) {
  var i = 0, found;

  for (; i < items.length; i++) {
    if (items[i].id === id) {
      return items[i];
    } else if (_.isArray(items[i].children)) {
      found = find(id, items[i].children);
      if (found) {
        return found;
      }
    }
  }
}

Update:

To find all matches - a slightly modified function (jsbin link above is updated):

function findAll(id, items) {
  var i = 0, found, result = [];

  for (; i < items.length; i++) {
    if (items[i].id === id) {
      result.push(items[i]);
    } else if (_.isArray(items[i].children)) {
      found = findAll(id, items[i].children);
      if (found.length) {
        result = result.concat(found);
      }
    }
  }

  return result;
}

Using lodash, you can do something like this:

_(data)
    .thru(function(coll) {
        return _.union(coll, _.map(coll, 'children') || []);
    })
    .flatten()
    .find({ id: 4 });

Here, thru() is used to initialize the wrapped value. It's returning the union of the original array, and the nested children. This array structure is then flattened using flatten(), so you can find() the item.

Another lodash option with children key and unlimited levels deep.

const flattenItems = (items, key) => {
    return items.reduce((flattenedItems, item) => {
        flattenedItems.push(item)
        if (Array.isArray(item[key])) {
            flattenedItems = flattenedItems.concat(flattenItems(item[key], key))
        }
        return flattenedItems
    }, [])
}

const item = find(flattenItems(items, 'children'), ['id', 4])