What are your teams' release processes like? How often do you push to product, who is responsible for push, etc.?

We suggest releasing weekly and have all related features push at once - then we conduct a manual group UAT (user acceptance testing) and update our release notes - while we still do the occasional hotfix this allow design and dev to talk through issues and not to force constant context switching for doers vs. managers

Pushes should occur on a certain day. It sets a good consistent goal for the team so people feel like something is wrong if you are not pushing something useful each week. 

We also suggest doing weekly product roadmap meetings to adjust priorities as needed and try to have a good handle on what is in the pipe for the next 4 sprints and then roughly the next quarter.