I would begin with an assessment as you would with any “new”client coming to you.
Be honest and direct without impeding the reputation of the other trainer.
Make it clear that you have your own style of training and go over your techniques and philosophies with them during the initial meeting.
As for keeping them on their current program, the length depends on their needs.
I switch things up all the time, but again this is between your scope of practice and the needs of your clients
Remember, everyone has their own techniques and personalities. Use yours to be the best possible trainer for your clients.