There should be no difference in behavior per browser. If the session from the 1st request was replicated successfully (this is done on the server side), then the failover node will be able to associate the requested session id with the replicated session with no problem. My hunch is that during your firefox test, the 1st node was killed before the session had a chance to replicate (replication is asynchronous by default). The fact that you saw different behavior from 2 different browsers was just a coincidence.
Thanks Paul for your reply, I captured packets during these tests and found that there was nothing unusual in both the cases. It turned out to be the Application's fault. Apparently the development team could not handle the session properly in the application.