In Exchange 2010 and earlier, public folder replication happens via SMTP, so the Transport service carries the replicated data. I believe but I'm not sure that the Information Store service is what generates and processes the replication messages.
In Exchange 2013 there is no public folder replication. Public folders are stored in a mailbox, which can be replicated in a DAG but there's nothing specific to public folders there.
Hi,
For Exchange 2010, the Exchange Information Store service controls replication related to the public folder hierarchy (the public folder tree), and the content of each folder.
Meanwhile, the synchronization process is email-based, replication-related email messages are sent from one Exchange server to another using the exact same protocols and connectors that any other email message would use.
More details about Public Folder Replication for Exchange 2010, for your reference:
https://technet.microsoft.com/en-us/library/bb629523%28v=exchg.141%29.aspx
For Exchange 2013, Incremental Change Synchronization (ICS) will be responsible for synchronization process with the public folder hierarchy. However, the content is stored in the public folder mailbox but isnt replicated across multiple public folders mailboxes. Please refer to: https://technet.microsoft.com/en-us/library/jj150538(v=exchg.150).aspx