Eventually, rolling your fingers through the shape will feel natural, just like fingering a G or C chord - which (at least for me, back when dinosaurs roamed the earth) feels impossible when I was first starting out. You want to mute with the meat of your palm on your picking hand, so as you sweep the strings you're immediately dampening the note you just played.
There's really (imo, keeping in mind that I'm a mediocre sweeper at best) two techniques that you need to practice separately. First is the actual motion of your picking hand, so that you get the chick-chick-chick-chick-switch_directions-chick-chick-chick-chick of the pick hitting the strings down. Accurate fretting doesn't help at all if you aren't sweeping the strings themselves. Then there's the actual fingering the notes in time with your picking hand. I think getting smooth with the actual sweep motion is harder to master than the actual fretting of the notes.
Edit: Check out my buddy Sean's YT page, he's a monster sweeper and has a bunch of good tips for people starting out.
http://www.youtube.com/watch?v=PiLYzTUSVUs