See screenshot, I used /locatebiome a few blocks from the present coordinates and got a biome at 324 ~ neg220, but then I moved a few blocks and the /locatebiome didn't pick up the biome at all when it should have.
Default world gen, no special settings. Latest 1.18 snapshot 21w40a.
I've also had it where moving just a few blocks locates another of the same biome but hundreds of blocks more distant.
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Height remains the same. If though you mean relative no matter what my height (I think that's what you mean), I was still only maybe a few dozen blocks away from the original reading, not enough to warrant it changing.
A drip stone cave is, maximally, only going to be a difference of, say, 128 blocks (the height of potential, give or take). I would need to then, at minimum, move 128 blocks horizontally AND already be on the threshold of it being too far away, which I believe is several thousand blocks, at least 3,000+. As you can see here, I was not.
I tested it a bit more. I think I understand now.
So whatever Y coordinate I am at it scans ONLY on that Y coordinate. So I could have dripstone caves right beneath my feet, but it ignores that because it's not reaching to the Y I'm at.
Is that how it currently works? If so, that's in a way beneficial, but mostly not...
This happens because the command only searches for biomes relative to your current height position, which is tracked in MC-206286 already.