Most of the time, it’s enough to copy the whole EFI partition to the new machine and update whatever boot entries are in there to point to the right new partitions.
In case of a switch to something like zfs, it’s a bit more involved and you need to boot a live Linux, chroot into the new “/” with /boot mounted and /dev, /proc, /sys bind mounted into the chroot.
Then you can run the distro-appropriate command to reinstall/ update grub into the EFI partition and they will usually take care of adding the right drivers.