initramfs error after kernel update on liveUSB

I'm using Linux Mint 17.3 with kernel 3.19. on a liveUSB with persistence (4GB casper-rw file).
If I update the kernel, after reboot I get initramfs error.
After searching the net, I understood the fact that after kernel update, initramfs and initrd must be updated in order to use the new kernel, but I didn't find a way to do it.
I found a nice piece of software called kernelUpdate (depends on lynx and xterm) which can update to kernel 4.4 instead of 4.2 present in Mint Updater.
After downloading and unpacking the DEB files, kernelUpdate gives me the same initramfs and initrd errors. Luckily, kernelUpdate doesn't make my system unbootable, like Mint Updater.
My question is how to edit/create new initramfs and initrd files to use the new kernel?

 

Thank Mutoid, I've read that page days ago while searching for the question in the title.
The USB install will wear & tear the USB in time, while liveUSB won't.
Another aspect of the liveUSB is that I can use it on multiple laptops/PCs without any problems.

P.S. There is a way, but I didn't find it...just yet.