FS v5.6 - Troubleshooting

I've configured an Epic-Story Listener and selected Sync on Link. When I update the configured fields on Epic, they are successfully copied to already linked stories but when I link the story to an Epic for the first time using the Epic Link field, the field values are not copied from Epic to Story.

You probably have the fields (that are configured for sync) present in the Edit screen (of the story issue) where you link your Story to the Epic.
If so, your field values are most likely copied successfully but then reverted back to the original value. (You can probably see this in issue history).
Long story short: Due to some design decisions made by Atlassian during development, Field Sync does copy the field value but Jira overwrites that value with the value in the Edit screen.
This issue is caused by a known Jira bug that can be tracked here: https://jira.atlassian.com/browse/JSWSERVER-20162
Voting for the Atlassian issue in the link above will tell Atlassian that you are affected by this problem and might speed up its fix by Atlassian.

You probably have the fields (that are configured for sync) present in the Edit screen where you link your issues.
If so, your field values are most likely copied successfully but then reverted back to the original value. (You can probably see this in issue history).
Long story short: Due to some design decisions made by Atlassian during development, Field Sync does copy the field value but Jira overwrites that value with the value in the Edit screen.
This issue is caused by a known Jira bug that can be tracked here: https://jira.atlassian.com/browse/JSWSERVER-20162
Voting for the Atlassian issue in the link above will tell Atlassian that you are affected by this problem and might speed up its fix by Atlassian.