1.) you could check on the device's settings see if it set to  "No Limit"

2.) If so, this could be a potential factor that is causing the slow sync of mails since this value allows ALL of user's emails from their mail database (which is too LARGE) to sync in to  the device. To fix this, have them set to this to something like 3 days or 1 week.

3.) Check mailfile size check if near the limit or over the limit
    - Domino Admin client > mail folder > select user's mail file > see if the Logical Size/Physical Mail file Size exceeds the Quota.        

   - or go to their iNotes/notes client and check the quota information (RED indicates overquota). 

4.) Check the distance and network between Traveler Server and user's mail server

5.) Do a trace or a ping to check latency in the network
trace <ipaddressOfYourTravelerServer>                                   
trace <hostname>                                                        
trace <servername>                                                      
trace <mailserverOfAffectedUser>                                        
.                                                                       
e.g.                                                                    
trace 12.345.678.90                                                     
trace http://ext-sampleserver.com/traveler                                                                     
trace homemailserver/SRV/MSC                                            
trace mailserver1/SRV/MSC  

I agree with Tristan. There can be a lot of factors why you are experiencing delays in sync. You can check if the messages that were sent have large attachment. For example, a huge image at the signature or something like that. 

This can also be a Network issue. You can test using either WiFi or cellular data to try and sync the device and see what's faster on top of checking the Mail server and Traveler server communication speeds. The ping speeds between these servers should generally be less than 50ms. Also try pinging with large packets to check the speed as well.