Alot of us have prepared for having to bug out and flee a disaster. We have our tools, our bags, and all sorts of other supplies. We have thought about which routes we will take and where we will go. What I've found, even in my own life, is what about our children, especially small children. Planning a bugout bag for a child is diffrent then what you or I would think of carrying on us. Me and my wife have come up with our thoughts on the matter and what we are doing for our family:

Backpack - Remember when picking out a bag it needs to be something your child can carry, or you'll be carrying it.
Clothes - I would pack a couple of sets for easy access, you don't want to be digging around when the baby needs a change of clothes.
Small toys - A few small sturdy toys for you child to play with is always a good thing to have. i.e. toy cars
Books - Have a few of your childrens favorite books on hand for them to read.
Diapers & Wipes - Self explantory. I'd pack more then you think you would need. Don't forget Diaper Rash cream. 
Snacks - Having a few of your childs favorite snacks on hand can be a quick morale booster

Bugging out with children presents it's own challenges and having a bag dedicated to them is a must
Remember children, especially little ones, are not going to understand whats going on when disaster strikes and your forced to bug out. Having items on hand like toys and books will help them to cope and will help to keep some sense of normalcy during a hectic and crazy time. Encourage them to express themsleves and their feelings in ways such as drawing pictures about how they feel. Also having things on hand like a few favorite toys and snacks will easily boost their morale and spirits allowing you to focus on the task at hand and keeping your family safe.

Something that we also recommend having on hand is an all-around entertainment bag with crayons (I use the triangle instead of the round so I do not have to chase down dropped crayons) or colored pencils depending on age, age appropriate coloring book, books, small toys (ex: mini slinky, toy cars), mini travel games, snacks and drinks. We use these when going to an appointment, ER visits, visiting someone in the hospital, hanging out at friend’s houses that don’t have kids toys and entertainment, etc.