Comment by jayd16

As the manager is accountable for the problem being solved, they are responsible for making sure processes are optimal.

There definitely should be an onus on individuals and teams to reflect and generate their own improvement actions to that end. Scrum Retros are a good example of this. In this case, the manager is responsible for process improvement by chairing the retro, ensuring that the team has the info needed, and has the space to implement actions. Scrum Masters chairing retros can be seen as a form of coaching.

There are also times when process improvement means directly stepping in and directing the team to do something differently. This can happen for lots of reasons; one example may be a manager taking over an existing team under fire and identifying immediate changes needed to dig them out. I've seen several teams with entrenched mindsets in this situation where process improvement is directed rather than discovered.

Ideally, the team drives it, while the manager is responsible for ensuring it happens successfully.

E.g. there is a big difference between "Why did we loose a day here, what can we learn?" vs "From now on each dev needs to review every pull request twice per-day". Might be the same ultimate action, but in the latter the manager is solving the issue directly.