In my province, the rent office arranges for two measures to counterbalance and increase mobility. Firstly, most rental agreements are made to end on July 1st - legally up to 1974 but now mostly out of tradition. This allows for much less friction, as everyone can change housing without any downtime with access to the full market.

Secondly, landlords are allowed to evict if they do sufficient improvements to the property, which allows a safety valve.

There are so many people that move in the month of July, that over 7% of the entire population - including people who don't rent - moves out on that day. It's pretty much impossible to get a rental truck to move for the entire month of July.

The statistics are that, around July 1st alone, 250 000 out of 900 000 people move. Add in people that don't move in this period, and you get almost half of renters in the entire city moving, despite strict rent control.

You just need to do your rent control carefully. There will be some reduction in mobility, but if you counterbalance them, you can have minimal impacts in mobility and large impact in affordability.