This type of information is typically stored in provenance data... You can use the SiteToSiteProvenanceReportingTask to get access to provenance events in JSON format and then filter the events to find the ones you are interested in. Each provenance event should have an event time which is the time the event was reports, as well as the lineage start time which is the time of the first event in the given lineage. So event time - lineage start time would be the time it took to get to current event.