There is a tendency to take much of the 4E rules and read it as literal. It is explicitly abstract.
In 4E, the "Grab" action, upon a successful "hit" imposes the "immobilized" condition.
The real question is "Can you immobilize a Swarm?". I think that is a definite yes, both by RAW and RAI. Can you then "Move a Grabbed Swarm?", this may be thornier, and where people have mentioned having a net or some such handy. I think you don't such extras. To require it runs the risk of the 10 foot pole problem of earlier editions (google it if you don't know). A foot is all you need (throw a bucketful of dice on the floor and move them with your foot - yup, that swarm can be moved) or a cloak for those pesky gnats, but even your flailing hands can do it.
I think it is well within the DMs right to impose a -2 penalty on such grab attempts (Invoking the DMs friend). I also think you should remeber that you don't have to move every creature in a swarm, just a significant portion of them. In many ways, a swarms aura is just more of the swarm, but not so many of them that you can target or prevent you from moving through.
The same sort of thinking can be applied to Prone and other conditions. In fact, my 4E claim to fame in our gaming group is the time I knocked a Black Dragon prone over a 1000 foot cliff (crashing, dmg 47). It failed its save every round to stop from falling and fell to its death! We described the prone effect as being a result of its wings getting blown out of position such that it fell. I was hoping to buy us a round or two of healing before it righted itself and came back...imagine my surprise as the DM kept rolling single digit save rolls all the way down.