Short answer, yes there is a solution to this. However, there are a few steps to make this possible:

1. Create either a .github/ or docs/ directory in your root directory.
2. Create a directory inside called PULL_REQUEST_TEMPLATE.
3. Create as many templates as you'd like within this directory, let's say template_one.md and template_two.md.
4. Begin making a pull request and append the URL with: "&template=template_one.md"

My example URL looks like: https://github.com/<org>/<repo-name>/compare/main...test-branch?template=template_one.md

It looks like the Pull Request doesn't keep track of changes to the target branch (I contacted GitHub support, and received a response on 18 Nov 2014 stating this is by design).

However, you can get it to show you the updated changes by doing the following:

http://githuburl/org/repo/compare/targetbranch...currentbranch

Replace githuburl, org, repo, targetbranch, and currentbranch as needed.

Or as hexsprite pointed out in his answer, you can also force it to update by clicking Edit on the PR and temporarily changing the base to a different branch and back again. This produces the warning:

Are you sure you want to change the base?

Some commits from the old base branch may be removed from the timeline, and old review comments may become outdated.

And will leave two log entries in the PR: