Update a local branch with the changes from a tracked remote branch

I have a local branch named 'my_local_branch', which tracks a remote branch origin/my_remote_branch.

Now, the remote branch has been updated, and I am on the 'my_local_branch' and want to pull in those changes. Should I just do:

git pull origin my_remote_branch:my_local_branch

Is this the correct way?

You have set the upstream of that branch

(see:

• "How do you make an existing git branch track a remote branch?" and
• "Git: Why do I need to do --set-upstream-to all the time?"
  )

git branch -f --track my_local_branch origin/my_remote_branch
# OR (if my_local_branch is currently checked out):
$ git branch --set-upstream-to my_local_branch origin/my_remote_branch

(git branch -f --track won't work if the branch is checked out: use the second command git branch --set-upstream-to instead, or you would get "fatal: Cannot force update the current branch.")

That means your branch is already configured with:

branch.my_local_branch.remote origin
branch.my_local_branch.merge my_remote_branch

Git already has all the necessary information.
In that case:

# if you weren't already on my_local_branch branch:
git checkout my_local_branch 
# then:
git pull

is enough.

If you hadn't establish that upstream branch relationship when it came to push your 'my_local_branch', then a simple git push -u origin my_local_branch:my_remote_branch would have been enough to push and set the upstream branch.
After that, for the subsequent pulls/pushes, git pull or git push would, again, have been enough.