Straight to the point, hope this is the right forum. I'm considering completing a great seller financing deal, but need your input on how I should manage the tenants already in place.

Back Story
Older gentleman wants to sell his investment property and we have come to an agreement on the terms and all signs point to us being able to close the deal soon. The issue: there are tenants in place that have been in the home for over 15 years and are paying $600/month, well below market rent. I do not think they are on any sort of lease agreement either, but will confirm this week. From what I've been told, they are great people and great tenants, but if I take possession of the house, I'd like to increase the rents to be somewhat closer to market rate. The seller has expressed wanting to give current tenants until February to move out. I want to be cognizant of everyone's situation, but I'm also not doing this for charity. How should I approach this situation?

My Thinking
Upon taking possession, I'd ask the current tenants to agree to/sign a month-to-month lease with monthly escalations until brought to just below market rate (the house needs a little work, but still habitable).

Assuming close by 12/31, I'd structure it as follows:
1. Agree to MTM lease with January rent payment being $800
2. February rent payment being $1000
3. $50/month escalation for each month after until reaching $1200/month

If they agree, great. If not, I can start renovations/repairs immediately and get top rent upon completion. My concern is them potentially damaging the property in retaliation for forcing them to move out. Is this a valid concern? How have you all managed this situation before?

Thanks for your help! Happy to provide any additional details if it helps the discussion.