I have actually done this myself in my demonstration environment. This requires you to use a FootPrints script called mrXMLConversion.pl in FootPrints\bin to dump the FootPrints database in SQL to XML so that it can be properly imported into the MySQL side of the house.
I suggest you contact tech support and request the migration documentation for Windows SQL to Linux MySQL. I want to note that you must be on the same version on both installs (Windows and Linux) and it's recommended to migrate from 11.1 to 11.1.
Let me know if you have any questions.
I've been waiting for the exact opposite (moving from Linux to Windows) for a bit over half a year now. The person who is working on it has had several setbacks with it along the way and there is an issue outstanding that is preventing this documentation being available. Since our installation is currently running 11.1, you might want to inquire about the 10.x documentation from support as that might be available.
The best way to deal with this if you can't upgrade your production environment to 11.1 is to:
1) install the same 10.x version you have now on your production environment (hopefully 10.0.2) onto the new server.
2) then perform migration steps to migrate your 10.x production install to the new server's 10.x install (with documentation provided by support).
3) Upgrade the new server's install to version 11.1 ***
*** Be advised, before installing/Upgrading to version 11.x on Unix/Linux, you must install Tomcat on the new server. Support can provide information on this.
There are incompatibility issues migrating directly from older versions into 11.x when changing databases.