We used to do a lot of split stays back when we were
DVC members. In those days, weekend points were so high that for our long trips that spanned two weekends we simply never had enough points to cover it! So we got a lot of practice.
It always worked best when we had a rental car.
We always packed a separate bag for that shorter stay, which for us was always on the front end. That way we didn't have to unpack everything.
We never planned any ADR's on "moving day" and in fact we often just enjoyed the "new" resort on the day we moved, hanging out at the pool etc. That way we weren't at a park wondering when our room would be ready.
We only did this when our kids were past napping age, frankly I would not want to do this with very young kids.
I think it can work and for sure do the moderate first. That way you aren't leaving a deluxe for a mod and feeling down about that!
If you don't have a car, I do suggest taking a cab. Disney will move your luggage for you, but snags happen. Once Boardwalk didn't move our luggage over to All Star Music and we checked in 8 or 9 hours after leaving BW to no luggage! BW found it but we never let Disney move our luggage again. We have cabbed our fair share around WDW and from any resort to any other resort have never paid more than $20 to cab between the two.
I also do think split stays are best for trips of a week or longer. In fact, ten night trips are my shortest limit for a split stay. If you are only going for say five nights, well to me it is not worth it. It does give you two different experiences, but it can chop up the trip and as Sammie said, your moving day can make you feel homeless for quite a few hours. And if the weather is bad or someone isn't feeling well, you will be missing your hotel room!
My favorite split stay was when we did three nights at the Poly followed by two weeks at OKW. Because we had such a long trip and a rental car it worked great. The time we did three resorts in 8 nights (shortage of points and adding another night on to save on airfare) was like Hell week and I wasn't sure I'd ever get dh to do a split stay again.
We no longer do split stays and its nice to check in and stay put. But under the right circumstances, I'd do a split stay again.