The kernel can be flashed separately in an android device therefore; it can be upgraded or downgraded depending on the ROM you have on the phone however, the kernel is very specific with ROMs. Every ROM with kernels specifically built for it.
How to Flash Kernel On Android
Mostly, custom recovery is required to flash a kernel except in the case of HTC devices with unlocked bootloader and S-Off whose kernels can be flashed using flashable zip files and Samsung phones whose kernel can be flashed using ODIN.How to Flash Kernel Via Custom Recovery
1- Download the specific kernel for the phone via Google, or if you have specific link to download from. Just make sure that the kernel is for the phone.
2- Put the downloaded kernel in the root folder of your SD Card or the phone's Mass Storage.
3- Reboot the phone to recovery using the right key combinations for the phone or by using the command line adb reboot recovery
4- From the recovery, wipe the cache + Delvik
5- From the recovery, choose Install ZIP then, select Choose ZIP from SD Card. Confirm the selection and wait for the process to complete and reboot the phone.
To confirm your new kernel go to Settings <<>> About Phone <<>> Baseband Version
The Kernel version is found somewhere around there usually; right after the baseband version.
Safety TIP:
Do always backup the whole content of your phone using a Custom Recovery before you attempt to flash any Kernel file on your phone.
In case anything goes wrong, you would be able to reverse all the actions.