- In analyze mode, i'm not sure.
- In game mode, analyzing from scratch i think is called "Brute Force" parameter.
MOSTLY/MOST engines are programmed to not analyze from scratch, but from a specific point of good moves. This allow engine to reach higher depths faster, which is better most of the time. This proved to make engines stronger in actual gameplay.
BUT obviously engines can skip some crucial moves sometimes, sometimes even lose a game because of that. BUT most of the time, skipping some proved better.
How engines are designed what to skip and what not, that's beyond me. 
After each move, does the engine analyse the position completely from scratch? Or does it refer to and continue on from all the analysis data it collected on its last turn?
If it depends on engines, then let's say Stockfish.