Players don't tend to switch clubs before their contracts run out, though it is possible if they're not wanted ir a compensation package is agreed. Teams can also take players on short-term contracts, basically loans, from other teams. That's not so unusual with non-international southern hemisphere players coming to the northern hemisphere as injury cover.
I think the playoffs make it difficult for Bristol to recruit as players would be less willing to commit to a side who might not be playing top flight rugby. If they survive this season then I'd expect them to bring in lots more players in the summer.
Coaches/directors of rugby don't often leave before their contracts are up or the end of the season either. I know that Holley was let go as head coach but no idea who has replaced him.