Do hot questions that get migrated stay hot in their newly found home, or does migration trigger a reset of the hotness calculation and an instant cooldown?
The hotness score is not tied to the question directly.
That score is calculated every few minutes to determine what questions will appear here, and it is calculated "from scratch" every time, not taking previous score as base for anything.
So when a question is migrated, when the hotness score recalculates, it might give high enough score to the question on its new site, if it has enough score/answers. But keep in mind that each site has a special "negative factor", which affects the chance of a question from that site to appear in the HNQ.
Bottom line: even if the question will have exact amount of score, answers, and answers score, it might not be "enough" when on the other site. But if it does, nothing else will stop it from appearing as HNQ.