As we all know that when we create a live copy of experience fragment or rollout any experience fragments the site's link reference inside it does not get updated as it gets updated in a page. This feature is not available OOTB.
In order to achieve that we can create a custom rollout config and can achieve our use case.
Add Synchronization Actions to the Rollout Configuration
Go to /apps/msm , create folder named rolloutconfig
Under this node create another node
jcr:primaryType ----- cq:RolloutConfig
Title: Update XF Site Links
Under this node now create another node
Implement LiveActionFactory :
The following LiveActionFactory class implements a LiveAction which updates the link refence in experience fragment on creating a live copy or rollout from master experience fragment. I have excluded countryselector component from update , you can update that part according to your needs.