Shooting run and gun style can be very challenging, even with the "smaller" DSLRs that many people use these days. I have found a combination that I've come to use almost every time I can't have the camera on a tripod.
It consists of:
Canon 5D Mark II - Pretty self explanatory.
Canon 24-70L Zoom Lens - Good zoom range for nearly everything you'd run in to and the f/2.8 is pretty decent for low light.
Small HD DP4 w/ Lens Hood - This is essential with its focus peaking feature. I use the hood as opposed to the viewfinder attachment when I'm running around because its lighter and I don't have to have the camera up to my face to see what I'm filming.
Monopod - I use a simply, light weight monopod just to stabilize the camera on. You can also use it as a sort of poor mans stedi-cam which can be helpful when you have to walk (or run) and film at the same time. Its also more comfortable when your waiting for shots or taking a breather to just rest the monopod on the ground, vs having to hold a shoulder rig all day long or finding somewhere to set it down without risking your gear.
Magic Lantern is a 3rd party firmware that can enable many features on your Canon DSLR that are not natively available out of the box. Everything from focus peaking, to zebras and even HDR video can be achieved with the use of this firmware.
Another handy feature is the firmware's ability to instantly tell you you camera's shutter count!
In order to do this, follow these easy steps...
1. Make sure you have the latest and greatest Magic Lantern installed.
2. Turn on your camera, and access the Magic Lantern menu.
3. Scroll to the "Debug" tab and you will see your "Shutter Count".
Your shutter count will be the total number of pictures taken + number of LiveView actuations.
The camera I'm using as a test for this has about 5,000 actuations. Pretty fresh!