It is possible to run the utility rpi-update (from here: https://github.com/Hexxeh/rpi-update) directly on RedSleeve Linux on a Raspberry Pi.
First you need to mount the boot partition in the running system. RedSleeve already has a /boot directory but it only contains an unused splash image.
mv /boot /boot-orig
mkdir /boot
Check your device names use:
fdisk -l
Mine were /dev/mmcblk0p1 and /dev/mmcblk0p2
mount /dev/mmcblk0p1 /boot
This should then allow you to see the true boot partition which will look like this:
ls -l /boot
total 34433
-rwxr-xr-x 1 root root 2047272 Jun 1 14:15 arm128_start.elf
-rwxr-xr-x 1 root root 2047272 Jun 1 14:15 arm192_start.elf
-rwxr-xr-x 1 root root 2047272 Jun 1 14:15 arm224_start.elf
-rwxr-xr-x 1 root root 273 Apr 19 08:58 boot_enable_ssh.rc
-rwxr-xr-x 1 root root 16528 Jun 1 14:15 bootcode.bin
-rwxr-xr-x 1 root root 124 Apr 19 08:58 cmdline.txt
-rwxr-xr-x 1 root root 26 Apr 19 08:58 issue.txt
-rwxr-xr-x 1 root root 4195844 Jun 1 14:15 kernel.img
-rwxr-xr-x 1 root root 6194612 Jun 1 14:15 kernel_debug.img
-rwxr-xr-x 1 root root 16344532 Jun 1 14:15 kernel_emergency.img
-rwxr-xr-x 1 root root 314691 Jun 1 14:15 loader.bin
-rwxr-xr-x 1 root root 2047272 Jun 1 14:16 start.elf
You can then edit your /etc/fstab if you wish to have this mounted each time you boot. My /etc/fstab looks like this:
/dev/mmcblk0p2 / ext4 noatime 1 1
/dev/mmcblk0p1 /boot vfat noatime 1 2
devpts /dev/pts devpts gid=5,mode=620 0 0
tmpfs /dev/shm tmpfs defaults 0 0
proc /proc proc defaults 0 0
sysfs /sys sysfs defaults 0 0
Now we need some items installed using yum:
yum install wget git diffutils
Now get the rpi-update util and make it executable:
wget http://goo.gl/1BOfJ -O /usr/bin/rpi-update
chmod +x /usr/bin/rpi-update
Now simply run:
rpi-update
Reboot and your on the latest RaspberryPi kernel code.