For smaller strollers that can fit into the X-ray machine, you have to fold them and put the on the conveyer belt. Larger strollers, like my single BOB get hand wanded. Depending on the individual, Security can be either be very friendly and courteous or really annoyed about manually checking your stroller, but they will do it.
I've flown on United, American, and Southwest in the last year, and have never been charged for gate checking my stroller or baggage checking my car seat (they don't count it as a piece of baggage). In about two dozen trips in the last three years, I've never had a stroller damaged when I gate checked it. (Not that it can't happen, of course.)
Unfortunately, you absolutely cannot leave the children in the stroller. Security, no matter how nice they may be, cannot even consider the request. (I've also had to take my cat out of his carrier.)
You must remove the child AND all extraneous items from the stroller. You can't have cups, packs of wipes, or any kiddie clutter in the stroller when they wand it, so make sure that all of your stuff is in bags or carriers that can be easily removed and put on the conveyor belt.
Also, they consider diaper cream a "liquid" and it will be confiscated (although I didn't know this, and didn't have my diaper cream confiscated until something like my 15th trip through airport security -- a fact I hope terrorists have not noted ;-)
That said, I always use my BOB at the airport now. I'm a single parent, so it's usually just my son and me when we travel. Initially, I thought smaller was better, and tried smaller umbrella strollers, but couldn't steer them one-handed and still pull luggage. And I can dump a lot of coats, carry-on, other stuff on the BOB. If it's you and your spouse, or another adult, and you don't have a ton of carry-ons, you should be fine with little umbrella strollers.
When my son was younger and prone to wander, I used his "teddy bear" harness to keep him near me when I was occupied with security or other business. (This was especially nerve-wracking in the Denver airport where there is an escalator directly behind security.) He was usually OK with that, since I used it sparingly, and let him hold the teddy bear's "tail" the rest of the time.
Good luck!