We have done split stays many times and have never had an issue. They can even keep refrigerated items for you between resorts.
What we do is one vof us (usually me) goes to the new resort early in the morning (if we don't have a car, as soon as the buses start running). By bus, just go to any open theme park and take the bus to the second hotel from there. This early morning trip is not scrictly necessary, but it will increase your chances of getting a room request and does get it out of the way (I usually do it while others are still getting up).
As mentioned, someone must wait with the bags for them to be picked up. They'll give you a receipt. Keep in mind things can get backed up, so take anything you need for the day (medicine, swiming suits, etc.) with you.
Then we generally go to the theme park nearest the new resort. We spend the day there and then go to the new hotel when the room is ready (or to check in if we didn't do that before).
To minimize waiting, get your bags ready well before the hotel's check out time (since many wait until the last minute!). Checking in very early in the morning is great (no lines until the first buses start rolling in from the airport). If you are a real night owl or early riser and have a car (or what to pay for a cab) you can even check in as early as 2 or 3AM I think (don't count on a room being ready until 4PM though!).
It is a pretty easy process and if you have a family with you, splitting up who checks in and who waits for the bags can make things faster (rather than having everyone wait for both).