Overlay disk space changes after a few reboot

Why does the disk space (df -k) on the overlay changes after a few reboot?
Does the boot sequence writes or check the overlay space?

What exactly changes in df -k ? Please provide the output before and after the reboots?

The disk usage on '/' /overlay goes up. It does not do this all the time.
I have also added to the top of the file /etc/rc.local to see if it was increasing after execution /etc/rc.local
df -kh > /tmp/disks.txt

It does not appear to increase. Both /tmp/disks.txt and df -kh display the same usage.

So in conclusion, the / disk usage increases every now and then before /etc/rc.local gets call.

'/' is the root of the device. It might change depending on the usage of the router. Maybe it is storing some data elsewhere that you dont know. Unless it changes significantly, for example use 100 mbs or something, there is no need to worry about it. But maybe you dont have a ExtRoot setup so your device has a limited memory so it might be a problem.

This means on every reboot, the os does a write which puts wear on the flash

Run the following...

du -s -h /*

/overlay is part of your root filesystem.

Could be some symlinks to deleted files.