If I can add a word of advice for our passengers - well, maybe two...
First, the obvious - try not to leave anything behind. Of course we're all human, so on to #2.
As soon as you discover that you've left something behind (either in the boarding area or your inbound connecting flt), let someone know. The sooner the search begins, the better the chances of finding it.
From my experience, if you mention it to the F/A's and give them enough information (where, what, when, name), they'll pass it on to us up front. When workload permits, we can either call back to the departure station (or contact them through whatever station we're over) and get the search started.
Obviously, we can't guarantee results. Over the years I've been able to pass the word back that the "lost" item will be arriving at the passengers destination on so-and-so flight at such-and-such time. Unfortunately, there have been the times that the best we could do was tell the passenger to talk to a CSA at the end of the flight.
Jim