When ever userA updates a JIRA issue, the JIRA issue automatically assigned UserA as the assignee
Create a mapping for JIRA Assignee and Salesforce CustomField type reference for contacts called "AssignedTo"
Associate a case (Let's call this Case1) with a JIRA issue (let's call this JIRA-1)
Configure in JIRA to auto-push (JIRA -> Salesforce)
In Case1, assign the value of "AssignedTo" to UserA
In JIRA-1, the assignee is UserB
Note: there is no user mapping for UserB, but there is for UserA
Update any field in JIRA and JIRA-1 will have UserA as the assignee (instead of retaining the assignee as UserB)