How? I think the bank will just sell the house, possibly for a lower price then the market value. Then, the previous owners need to rent somewhere. As there are less people able to pay a mortgage, the more people need to rent.
As seemingly everywhere around the globe interest rates have gone up, the influx of new home buyers have gone down. Those people have to compete with each other at the rental market. Unless your country is able to create a new big supply of rental homes to somewhat balance the market, the rental price is definitely not stable, but trending upwards.
In the Netherlands we are getting stricter rental controls to keep the rental price to not increase that much. Some people are sitting between a rock and a hard place now: not able to buy, not able to save due to high rents.
Rental prices might fall if people are forced to sell their houses at lower prices. In my area, people buy houses on 30-year mortgages with variable rates and require an in-law suite to finance it partially. The margins are so thin that I encountered people who had to sell their homes because of increased property taxes.