Similar to upgrades, the idea of an install of updating a Splunk install for a new architecture or new operating system is to just unpack the appropriate splunk version on top of the existing install.
Migrating a splunk install to a new operating system builds on this, with some caveats. The general idea is as follows.
1. Shut down your old Splunk install: splunk stop
2. Copy (via whatever means, scp, rsync, tar, zip) the Splunk install directory to the desired install location on the new computer.
3. Unpack or install the Splunk tar or package for new system on top of the copied files.
4. Start it up. splunk start
You can even "upgrade" to the same version you were already using with this method, just to validate that the operating system transition was effective.