alt Hacker NewsGitHub pull request pushes you a notification/e-mail to handle the merge, and you have to handle the pull request mostly online.
I don't know how you can download the pull request as a set of patches and work offline, but you have to open a branch, merge the PR to that branch, test the things and merge that branch to relevant one.
Or you have to download the forked repository, do your tests to see the change is relevant/stable whatnot and if it works, you can then merge the PR.
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edit: Looks like you can get the PR as a patch or diff, and is trivial, but you have to be online again to get it that way. So, getting your mails from your box is not enough, you have to get every PR as a diff, with a tool or manually. Then you have to organize them. e-mails are much more unified and simple way to handle all this.
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In either case, reviewing the changes is not possible when you're offline, plus the pings of the PRs is distracting, if your project is popular.
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You could set up a script that lives in the cloud (so you don't have to), receives PRs through webhooks, fetches any associated diff, and stores them in S3 for you to download later.
Maybe another script to download them all at once, and apply each diff to its own own branch automatically.
Almost everything about git and github/gitlab/etc. can be scripted. You don't have to do anything on their website if you're willing to pipe some text around the old way.
Seems like you found it, but for others: one of the easiest ways to get a PR's diff/patch is to just put .diff or .patch at the end of its URL. I use this all the time!
Random PR example, https://github.com/microsoft/vscode/pull/280106 has a diff at https://github.com/microsoft/vscode/pull/280106.diff
Another thing that surprises some is that GitHub's forks are actually just "magic" branches. I.e the commits on a fork exist in the original repo: https://github.com/microsoft/vscode/commit/8fc3d909ad0f90561...