The kernel will usually be not that important - more often you will find that your older Slackware installation has older libraries than your newer system. Since most software is dynamically linked, chances are high that packages will break if run on the newer machine then - your mileage may vary of course, depending on, well, the dependencies of the specific apps you are compiling and if they got updated between the two Slackware releases.
I would suggest to install something like VirtualBox on your older system and to create a virtual machine with Slackware 13.1 or what you like inside that. This way you can run your old system and compile software for your new systems.
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