When miners try to find the next block, do they always include new transactions in the block if they have a high enough fee? Or do they sort of collect a bunch of transactions and then proceed to search for the block while neglecting new incoming transactions?

So mathematically it does not make a difference for them right? They will always add whatever comes in if there is still space in the block and if the new transactions have a fee? My thinking was that they then would have to "start over" to find the right hash as adding new transactions will completely change the hash of the block. But that is incorrect, right? It does not really matter, as the probability of finding the block is as likely as before.

Thanks in advance.