@Sergio I've looked around for this periodically over the years, and while it would be a significant win for linux accessibility to my knowledge it doesn't exist. I've never seen an installer that so much as volunteered to help with a backup, much less offered migration/restore afterwards. All the linux installers I've ever touched believe that backups are the user's pre-work responsibility and that they're going to have permission to pave the entire drive flat.
@mhoye @Sergio I wouldn't think the makers of a distro would want the responsibility of ensuring people's files are transferred safely. To much to go wrong. Nothing gained.
It would need sufficient space on the USB installer to hold all the stuff or make a partition to hold it? And it would have to ask what folders you wanted copied.
All the user has to do is drag Documents, Photos, Videos, etc to an external drive. If a person can't do that, they don't have any business installing an OS.