I use a small cooler bag with an ice brick in it. I bought really skinny ice bricks from the supermarket so there is plenty of room. I pack all DD food into small bowls with lids and or Tupperware or Heinz Baby Basic snack containers. Most shopping centers have parent rooms with microwaves so you can heat food if required. If I'm eating at a cafe, I ask them to zap DDs food (if required) for me.

I make a lot of DDs food but have always found Rafferty's Friend sachets excellent. No need to keep them cold, unless they are open, and my DD is happy to eat them at room temperature so they've been great when I don't have access to a microwave.

HTH