The best behaviour, imo, is having the search query bound to a folder. So, if I leave focus on another folder, the search is not applied. If I return to the original folder, the search & results are restored.
Another way to handle search is how Thunderbird does it. Each search/results opens in it's own tab. From there you can drill down on a timeline to pinpoints the email you're looking for, like such (note, the time-line is above a paged list of email below it that correspond to the time-frame of the search):