Travis cannot do anything with the source, Github is in control of it. Github is also the one who creates “preview” merge commits that Travis uses in PR checks.
You can use a Github bot to automate the task. E.g. https://github.com/travis-ci/travis-ci/issues/1620#issuecomment-406123643 suggests https://github.com/tibdex/autorebase that “auto-rebases” branches, and I also found https://github.com/bobvanderlinden/probot-auto-merge that can “auto update” them (dunno what specifically each means).