I would let them know that their current lease is ending in 30 days and their new lease (your lease) is starting then. I purchased a 6 plex unit a couple years ago and it was fairly smooth transferring over leases. With my new lease I did a rent increase in another 2 months. I wanted to give them time to find a new place if they desired to. I also made a regular presence at the apartment to try to let me know that I cared about it. Painting exterior doors, clean up the lawn, minor landscaping. Arguably not "need to do" items; however it shows that you care, which makes them more willing to pay higher rents.
You won't need to pay to have the current tenants move. One thing to consider is that they're currently paying under market rent. Fair assumption their apartment conditions are below nice & shiny fair market rental today? The few thousand it may cost to increase each unit up to nice & shiny fair market rent, may be years to get paid back. I've gradually bumped my rents up so that there is enough leftover cash flow to fund any future turnovers. Food for thought, do what works for you. Enjoy and congrats