The connector will trigger updates to Salesforce Object automatically because the Listener in the connector is enable by default base on mapping fields.
This is a list of events captured by Listener and trigger "Push Change to Remote System" function:
Issue Updated
Issue Assigned
Issue Resolved
Issue Closed
Issue Worklog Updated
Issue Worklog Deleted
Issue Work Logged
Issue Reopened
Issue Started
Issue Stopped
Issue Generic Event
Custom Event
In other words, it will firing an updates after JIRA issue being edited and after a workflow transitions.