First and foremost, I would suggest making sure that your children can, and will, perform regular chores every day without fail.
If you don't already have a pet, then you can establish some other daily chore as a surrogate. If they're not already doing such a chore, you can assign a new one, and explain that you're doing so to see how they would handle the responsibility of a pet.
Make sure they do the chore every single day. I'd suggest waiting at least a month, to make sure they don't lose interest over time, and you should probably avoid establishing a clear timeframe with them (its much easier to keep up that chore in the third week knowing you only have 7 more days until you get that pet you wanted!).
You can also expose them to other aspects of taking car of pets by finding friends who have a pet of the type you are considering, and arranging for your children to help out with some of the less fun tasks. Walking someone else's dog (supervised), and then being responsible for picking up their droppings is a big test of a child's commitment, as would scooping litter for a cat, or cleaning out a rabbit's cage, or cleaning a fish tank.
However, since it sounds like you already have a dog, then the best demonstration is that they simply take on the responsibility of feeding and walking your dog. Its fine if you supervise, particularly if they aren't physically big enough to safely walk your dog, but you should put them in charge of picking up the droppings and disposing of them safely. If they can't shoulder a portion of the burden of caring for a pet already in the household, they won't be ready for their own pet.