It can take up to a month to get good autotrophic migration from old media to new media. Migration is greatly reduced if the media are not -inside- the same filter together, but there -is- a tiny benefit to having old media just in a tank as opposed to not having it at all.
Ethan is right, 95% of what we're trying to accomplish with mature media (MM) transfers has to do with the sometimes very messy process of figuring out how we can cut up, divide or otherwise somehow separate the MM, in whatever form it was in for the old filter, into some new shape that we can then place into a pocket of space within a media position of a new filter. Often the media of the new filter must be cut or othewise damaged to make the space available.
If you are "borrowing" the MM and there are still fish dependent on it in the old tank, its a better technique to simply "clean" the MM in the new tank and let the new filter begin sucking up the debris that it spews out into the water. Beginners often have a hard time making themselves do this in their nice sparkling new tank but the more muck you can get in there the better - the filter will eventually pick it up and you will eventually gravel vac the rest out after the fishless cycle is over and it will have a good chance of speeding up your fishless cycle.
Also, if there are fish in the old tank dependent, you should try not to borrow more than 1/3 of their main biomedia, leaving 2/3 to support the fish and to re-colonize the 1/3 new media material that replaces the borrowed media. Not only that but the maturity level of the colonies is a factor too. Any filter less than a year old is not really mature and should be treated with increasing delicacy (meaning take less of it or generally try to disturb it less) the younger it is.
~~waterdrop~~