I did this just over 3 years ago. I worked Monday to Friday evenings and also Saturday mornings doing house calls. Apart from a few elderly people who I visited during the day, everybody started coming to me. I didn't give them an option. I did this just after Easter and did lose around 8 pupils by the end of the summer term who I managed to replace quickly enough. The majority of my pupils at the time were below grade 2 standard, and I think I would have lost most of them eventually anyway, but having to come to me made them think about their commitment to continuing lessons.
I didn't reduce my fees but didn't increase them for 2 years either.
I built an extension to teach from and most pupils and their parents knew what was happening for almost a year before I started working from home. I'd suggest giving plenty of notice of your intention to be home based.
Overall, it's much better for me. I have more teaching time (no travel time between lessons - travelling 1 mile in my town can often take 10-15 minutes) and I'm much less tired by the end of the evening.