Just tried this at length yesterday on a usb install with persistence trying to upgrade from 4.13 to 4.15 kernel.
It looks as though you can update/upgrade anything on a persistent live usb ... except the kernel ... you can download the kernel image but cannot update grub to boot the new image as it's config file is on a read only partition.
I found a couple of methods of trying to write to the usb read only partition from a different identical OS but couldn't get any access to the read only partition that way either ... so it looks like there is no easy way of updating the kernel version on a persistent install ...