Jenkins Testing PR's - allowing broken builds
On GitHub - you can't. Basically, their PRs are temporary branches, and there's no check if base commit is HEAD of branch to merge in.
This problem is solved with gating systems like Zuul, which puts all PRs into queue to run tests.

You are talking about systems for managing and storing code, while the issue is in CI scope (i.e. Gitlab flow has this issue described in "Testing before merging" paragraph), Zuul is made exactly to fix this problem, because the amount of merges to OpenStack codebase per day is large (year ago they had 6 zuul instances to run tests for merges).