Here's what I do:
Add Laravel's official repo as a remote.
git remote add -f laravel https://github.com/laravel/laravel.git
View the files that are different between your project branch and the latest stable Laravel release within the release branch you're using for your project. The first column in the resulting list is a letter representing the change status of the file. A = Add, D = Delete, M = Modify, etc. Run git help diff and search for --diff-filter to see all possible statuses.
git diff --name-status <local branch> laravel/5.3
Check out the files from the laravel remote that have been added/modified/renamed, etc.
git checkout laravel/5.3 <file path>
Delete any files that have been removed from the laravel remote.
git rm <file path>
Update Composer packages
composer update
Update Node packages
npm update
Commit all of the changes
git add .
git commit -m "Update Laravel to 5.3.31'