I don't think they would allow that. I always drive down so have a car with me. Depending on my mood and where I'm moving to, I either go ahead and move my car to resort #2 in the morning and store my luggage with the resort or just put it in my car and leave it at resort #1 until I'm ready to move to resort #2. I have never had anyone bother it in the car.
They say they will deliver your bags by 4 but I wouldn't count on it. I've seen others say it has been after 5 before they got theirs.
To answer OP: Typically I go to a park on move day. I tend to stay in a value one or two nights at the front of my trip then move to an Epcot resort the rest of the stay (lately Dolphin). I use the value nights to go to AK and MK since you have to bus to AK no matter where you stay and unless you are staying at a Bay Lake resort, are going to have to bus to MK. Bonus for me, Dolphin drops you off at TTC at MK so I prefer to knock it out while I have Disney buses. I always have a car so I just drive over to resort #2 after I've had breakfast, I'm an early riser even on vacation. I go ahead and go to the front desk just to let them know I'm there, I've noticed they tend to try to get your room ready earlier if they know are there. I also never pick any special room, just first available, I don't care about view. I always have PHs (right now an AP) so if the park I go to is too busy at 2 I hop somewhere else. Since I stay at Epcot resorts the second leg of my trip Epcot is usually my go to move day park but since it doesn't open until 10 or 11 now days I've switched that to HS. I can waste enough time until 2 there then walk over to Epcot if my room isn't ready.