LMS does not handle these automatically, but you could use Netconfig to automate the backup. You could build a Netconfig job that copies the license files off to an FTP server, then deploy that job across all your ISRs. This will work in LMS 3.2 and 4.0.
You will need to create a user-defined template in Netconfig or just use the Adhoc template. The job needs to run in Enable mode (vs. Config mode). The FTP command will be interactive, so you'll need to use something like:
copy flash:/file.lic ftp://username:pass@10.1.1.1