Why are Stockfish evaluations relatively unstable move to move?

I'm reading about openings and trying to check the lines various sources give using the Stockfish engine built into the analysis board here.

I've noticed some instability in the way it evaluates positions. For example, I'll put some position in the open Sicilian in (perhaps 10 moves in say) and its evaluation after thinking for a minute will give completely different top moves than the initial evaluation, and for a single move I get evaluation swings as large as .3 pawns after this time. This seems reasonable but is a little annoying.

More interestingly, there are large swings in evaluation between moves. For example, suppose I put in a position and let it think for a minute, and the top move has evaluation 0.0. Then I will make that move, and the evaluation immediately swings, and this swing can be as large as half a pawn either way. Crucially, this evaluation change seems to remain even after I give the engine extended thinking time.

So, to recap: On move 10, the position will be at 0.0 according to Stockfish, and on move 11 it will -0.4 after making the top move. This seems wrong. Shouldn't Stockfish notice on move 10 that the position is actually -0.4?

I was wondering if anyone else has had similar problems, or if there is a way to eliminate it. Maybe I could just use Stockfish on my computer? In this case, an additional question: the Stockfish engine on lichess is being run in my browser, so switching to the desktop version wouldn't make it lose any strength, since the same computer (my laptop) is running both, right?

It is normal for evaluations to vary. For example if you look at the TCEC engine competition, you see how the evaluations of both engines at long thinking time vary from one another and from move to move. Especially in the opening the engines are helpless without their opening book. In the middle game the engines become stronger and the evaluations more accurate.

Thanks for the reply.

Do you think I would get better performance running Stockfish on my computer than in my browser? This is somewhat inconvenient, but if I really want the "truth," it seems like it might be a better bet. But I don't know anything about the technical details, which is why I ask.

Checking the nodes/sec measurement shows they're about the same. Which I guess is to be expected.

edit: actually, I get 3000 kN/s on my laptop using the maximal number of threads, and about 1800 in my browser. In case anyone else is interested in this.

I run SF from my PC, it is about 1500 kN/sec. On browser it is 150 to 350 kN/sec.
I prefer my computer.

Have you tried increasing the number of threads and memory available to Stockfish in your browser? It may help your performance.