I meet with all stakeholders and ask them: what worked well for acquisition; what could be better; what is one thing we can fix for our next acquisition as a means for process improvement.
I created a post-integration survey and send it in the end of the PAI process to all acquired team members. It has some specific questions on which integration aspects should be improved. Then I review the results and create Integration lessons learned which are then used to improve my PAI playbook/project plan.