For example if a local machine is given an arbitrary IP address from the DHCP pool and then I select "Always use this IP address" then there is no problem.
But the only way to force my miners to get IP addresses in a row is by setting them all static. Unfortunately the router decides to change that later on for no reason

Sounds quirky.. I simply setup static DHCP assignments in my router as others have suggested, it's how most of my networks are setup.