The people above me gave you great pointers. I just have a few things to add or reiterate, and I'll use this mix of mine as an example of both compositional and textural transitions.
If you already know what the sound design in section B is going to be like, in transitioning from A to B, you could use some tonal instruments from B as a lead-in component. I've been trying to do this all the time now, and you can see it many times here. EX: 0:58 - 1:00, 1:13 - 1:15, 2:36 - 2:37, etc.
Something as simple as a reverse can connect the dynamics (or at least, contribute to it). Experiment with reversing things like cymbals, white noise hits, fingerbells, and so on. I don't think I've had a good mix where I haven't done this... but this usually needs other accompanying techniques. EX: 0:28 - 0:30, 1:29 - 1:31, etc. Pretty much everywhere in this mix!
You can explicitly write a part that introduces the rhythmic contrast that you'll write in the next section. EX: 1:55 - 1:59
Drums should always be considered as a signaling tool, particularly if you go from a section with no drums to heavy drums or vice versa. EX: 2:25 - 2:26, 2:37 - 2:38
Try to match up the frequency spectra of the two sections. This is kinda difficult to pull off, and is not necessarily the first thing you think of, but sometimes you're not sure why it's not quite working, and there's just a small disconnect in the frequency spectrum that works out when it's all matched up. EX: 1:29 - 1:30, 1:57 - 1:59
(1:29 in particular has something interesting; the drone is swapped out for a sustained bass with almost no attack. They have similar frequency spectra, which can make it feel like nothing actually changed in the bass... and that's what I actually wanted!)