You can use the following flags --single-branch && --depth to download the specific branch and to limit the amount of history which will be downloaded.
You will clone the repo from a certain point in time and only for the given branch
git clone -b <branch> --single-branch <url> --depth <number of commits>
–[no-]single-branch
Clone only the history leading to the tip of a single branch,
either specified by the--branchoption or the primary branch remote’sHEADpoints at.Further fetches into the resulting repository will only update the
remote-trackingbranch for the branch this option was used for the initial cloning. If the HEAD at the remote did not point at any branch when--single-branchclone was made, no remote-tracking branch is created.
–depth
Create a shallow clone with a history truncated to the specified number of commits