I coach T&T, so our competitive progressions are a bit different from WAG/MAGs.
If they can't hold themselves up in a bridge kickover, we're not doing backhandsprings. I also want them to be able to do a spotted backbend. We work bridge kickovers & backhandsprings at about the same time, bc shoulder flexibility can make doing a back walkover on flat impossible, while a backhandspring is totally feasible.
We work front limbers before we even think about front handsprings. If they can't get their feet together before putting them down, they're just not ready (and there's nothing like spotting a front flopover that has momentum behind it. oh my).
They all learn front tucks on trampoline/off double mini well before on floor, but if they cannot do a front roll off 2 feet & stand up with their hands in front of them, they aren't ready. I don't think this is setting an unreasonable bar =p
For back tucks, again, first they learn it on trampoline or off the double mini (spotting off the end of double mini is nice...it puts them at such a convenient spotting height...). They absolutely MUST be able to do a back roll to their feet every time. The knee landing habit, ugh. And I'd like them to have a backdrop pullover (& so would USAG T&T) but will compromise depending on the kid. On tumbling, they need a good solid roundoff several backhandsprings, & I want them to show good mechanics in a standing back tuck before we add it to anything.
And I'm way picky about roundoffs. If it's not going to make the backhandspring easier instead of harder, we're not doing roundoff backhandsprings yet. Backhandspring series, sure. But not cartwheel pikedown backhandspring.