Automatically pull changes from the main branch to a number of protected branches

I have a repository with many protected branches. The main protected branch is A, to which code and updates are merged from developer branches; other protected branches are B (customized for one client) and C (customized for another client). Is there a way to configure updates merged in branch A to be automatically pulled into client branches B and C and trigger CI pipelines to run?

In the meantime, i do it manually on one of our servers via the command line, like this:
#For branch B

$git checkout B
$git pull origin A --no-edit
$git push

#For branch C

$git checkout C
$git pull origin A --no-edit
$git push

Thanks.