Perhaps this will help, Ryan: Remember that "up and down" involve movements of the flexion-extension movement pool (that is, the "pressure-control skill"), while edge change is a tipping movement (edging skill). While we may often rise while we reduce edge angles to release and start a new turn, the two movements (extending and tipping/flattening) are independent of each other. Rising does not necessarily release or change your edges, and changing your edges does not necessarily require rising.
The instruction to "rise to release and initiate a turn" is one of the more unfortunate, but most common, bits of poor advice out there. First, as I mentioned, it is not rising that releases edges, but flattening. Second, either way, it does not happen at the initiation of the turn. Initiation occurs once the edges have released--not when we START flattening (or rising). In other words, if you rise to start a turn, even if the rising accompanies flattening of the skis, the turn will begin not when you begin rising, but when you HAVE RISEN! If you want your turns to be smoothly linked, the rising/flattening must occur at the END of the PREVIOUS turn, not the beginning of the next turn.
This misinformation is related to the common mis-advice to "start a turn tall, then flex progressively to end the turn low." Think about it--if you want your turns to be seamlessly linked, they must begin and end in the same attitude and stance. I call this stance "neutral," which I'll define simply as the position from which the turn starts--whatever that may be. Clearly, the actual turn begins with a release of the edge(s), so "neutral" represents the moment the skis become flat enough to release their edges (which does not necessariy mean totally flat on the snow--neutral implies minimum "critical edge angle," a concept we have discussed here several times). Because the tipping/flattening and flexing/extending movement pools are independent, "neutral" can mean tall, short, or anything in between, depending on skier intent and conditions (in moguls it generally happens when the skier is flexed very low).
So--rather than thinking of an "up motion with edge change," think of FINISHING your turns in "neutral." If you do it right, your new turns will feel effortless, because everything you have to do to start them you'll have already done by the end of the previous turn. You'll already be in the right "position," and all necessary movements (i.e. the "crossover") will already be in motion. Your turns will be seamless, effortless, and incredibly smoothly linked.
No matter how much edge angle you develop in the middle of the turn, then, it should all vanish by the turn's finish. The turn should END with "critical edge angle"--any more and the turn is really not over, and you are not nearly ready to begin the next one. If you choose to rise (extend) as you flatten the skis, you should be at your tallest at the END of the turn, leaving absolutely no reason to extend to start the next one.
In summary, it is entirely unnecessary, although sometimes desirable, to rise/extend in the transition between turns. It IS necessary to release the edges in the transition, but that flattening movement does not necessarily require rising. I rise (sometimes) because I CAN--not because I HAVE TO. Rising gives me a moment to relax and rest my legs briefly--an advantage when free skiing for fun. It may not be the quickest way to release and reengage the edges, so I may not do it when racing. Because my transition in moguls usually occurs on top of a bump, I MUST learn to release my edges while remaining highly flexed.
Finally, all this discussion assumes offensive turns with no redirection of the skis during the transition. If I need braking (skidding) or otherwise want to redirect my skis before re-engaging my edges, I may well rise to "unweight" them, making the redirection easier. But that's another story....