Mary Poppins and Bert typically "stroll" in the mornings. We have found them walking around town square, and also wandering around Fantasyland. They don't stop and let a line form, but they will let you walk with them, talk to them, sign autographs while they walk and slightly pause for a picture or to let you get in front of them and take your child's picture while the child walks with them. Typically it is fairly easy to walk to the side or behind and get a turn with them, but one year there was a big group of adults that was basically shoving in front of the kids who were walking waiting to see them. That time Mary Poppins and I had to get a little more aggressive to get DD to her.
(Mary realized DD had been waiting and adults kept shoving in front of her and used her orderly personality to make others let DD have her turn as I slightly pummeled my way around the adults- as the others had done to us)
The Mad Hatter and Alice often stroll through Fantasyland and joke with guests and hold impromptu tea parties. Again, they don't often stand still for autographs and photos and let a line form, but they will sign autographs as they move and slightly pause for pictures. We have always gotten their autographs and photos.
Around the castle in the afternoons, characters will often pop up. You can also find random characters popping up in town square or right inside the park gates. Short lines will form for them. Our last trip, Captain Hook was strolling through Fantasyland. He took DD to a spot for photos, but then decided he didn't like that spot and moved somewhere else to play with her and take photos. There was no one around but he and DD. After we said goodby and left, a short line formed for him after people realized he was standing there waiting for people to come see him.
When you see a character, just follow. It is fine to approach them to talk or ask for an autograph or photograph- just be polite and follow their lead. If you follow them you will quickly figure out whether they are heading to a spot for photos, leaving, or strolling. Once the stepsisters and Cinderella walked past us (DD's favorites)- with the stepsisters bickering the whole way and Cinderella just walking and nodding to those as she passed. We followed them, and they stopped at wishing well by the castle and allowed a line to form for photos and autographs and we were first!
The big difference is that a lot of these characters, don't have handlers with them- so it feels like they are just out and about, which is nice.
We love that we never know who we might run into.
As far as the character meals, Minnie's breakfast is great. It is better than WDW characters meals. There are some characters that are almost always there, but you never know who else you might run into. I don't think we have ever seen less than 7 characters during that meal and once we saw 12. Goofy's used to be almost the same way, but in the last year they standardized that one, so now it is just like a WDW character meal.
Have a great trip! We think DLR/CA has a slightly more laid back feel, and we love that everything is so close together and quick walking distance. Be careful though-- we planned one trip to DLR and now we alternate coasts! I hope you love it as much as we do!
They sat with a few other guests and drank out of stuffed tea cups and then played a game, then pictures with each guest. So cute and magical! 
