A few months ago I wrote this about the clarinet:
If breath support, embouchure, and voicing are correctly established, then Crossing the Dreaded Break ceases to be a Thing. It’s just another note: a moment ago you were playing B-flat, and now you are playing B-natural. As long as your fingers get where they are supposed to go, then that’s all there is to it. Personally, I don’t even use the word “break” with a beginning student—there’s no need to get them all uptight about what really is a non-event.
My point was that crossing a register break is merely a fingering issue, and shouldn’t be turned into a big to-do about embouchures and equipment purchases and so forth. And I stand by that, but there is something I glossed over a bit that perhaps ought to be revisited in more detail, and that applies to register break crossings on all woodwind instruments.
The point that I want to return to is that of breath support. If it, and some other basic tone-production matters, are “correctly established,” then break-crossing is indeed nothing more than a new fingering or two. But assuming that breath support is 100% correct with a student just reaching the break-crossing stage is often a mistake.
Each note on the clarinet (and on any woodwind) has a certain level of resistance—that is to say, it requires a certain amount of air pressure to get the air column vibrating. Some notes are more resistant, and some are less resistant. As a sort of general oversimplification, we might assume that a long-tube note (with more toneholes closed) is more resistant than a short-tube note (with more toneholes open). Other factors do apply, of course: the size of the toneholes, whether the fingering is a “forked” fingering, and more, but let’s isolate tube length for the moment. So for the clarinet, having a break between A-sharp and B, we would expect to see this kind of resistance change while crossing the break:

(Note that the bar graphs here are strictly illustrative and not based on any real measurements.)
A beginner who is accustomed to the lower resistance of a few chalumeau-register notes might have intuitively developed just enough breath support to make those notes respond. When he or she attempts to cross the break, the breath support isn’t enough to overcome the increased resistance: