For Mercurial, this boils down to pull/merge/push in case of a named branch or, for topics, pull/rebase/publish/push or pull/merge in target branch/publish/push.
Here are the Git instructions from standard GitLab:
Step 1. Fetch and check out the branch for this merge request
git fetch origin git checkout -b topic/extradoc/bitbucket-pr-1 origin/topic/extradoc/bitbucket-pr-1
Step 2. Review the changes locally
Step 3. Merge the branch and fix any conflicts that come up
git checkout branch/extradoc git merge --no-ff topic/extradoc/bitbucket-pr-1
Step 4. Push the result of the merge to GitLab
git push origin branch/extradoc
Note that pushing to GitLab requires write access to this repository.
Tip: You can also checkout merge requests locally by following these guidelines.