Sorry you are having trouble and thanks for your patience.
Can you supply logs of the failing situation using the tool we provide in
http://www.displaylink.com/support/ticket.php?id=269 it will help us to diagnose the problem.
Does it work if you restart the DisplayLinkManager service? (not a solution but will help to narrow down the problem).
I'm not sure we've had any E7240s in house, but we've done extensive testing on Intel Haswell, and Dell have not reported an issue like this to us (they resell many products using our technology), but I hope we can get to the bottom of this.
Dan.