If I was in your situation, my plans would revolve around "bugging in" as my first choice. That would put less strain on the family and you wouldn't have to worry about the kids "pulling their own weight" or how long it will take to travel with kids and a dog.
But, as you know, you might not have any choice in the matter. In a Katrina like situation if your house becomes unliveable (floodwater or tornado damage or whatever) you might have no choice but to evacuate. So, you do need a "bug out plan," even if it's not your primary plan.
The first thing I would do is repack your critical bug-out supplies into packs or bags. While you'll obviously want to bug out in the mini van if you can, you need to have the gear packed so you can continue on foot if needed or just leave on foot from the start.
What if the vehicle runs out of gas when you are stuck in mass evac traffic? Or what if it's destroyed by a falling tree in your driveway before you even try to leave? Or what if the roads are unpassable by vehicle due to debris, etc? You get the idea.
In any situation where you have to abandon the vehicle or just can't take it in the first place, you aren't going to be able get very far trying to carry things in those plastic storage boxes.
What I would do is prioritize your bug out supplies. Pack your most essential items in the packs you are your wife will carry. Give your kids some good kid packs and have them carry some extra water and maybe a little food. Water is going to be most important so the more, the better.
Once you have the packs packed for you and your wife, and small packs for the kids, repack the less essential stuff into bags that you could carry in addition to your packs. You'd probably only want to go a short distance with these extra bags, but if you make it close to your destination before you have to go on foot, you might want to have those extra supplies and the more convenient they are to carry, the better.
Finally, since Plan A is to take the Mini Van, don't be afraid to load it up with more then you could carry on your back, even if it's in those plastic boxes. Use the consumables (food, water) in the van first and save the stuff in the packs for last.
When you are bugging out with your kids, try to find room for some comfort items or fun items. A favorite toy or blanket or what not. If you can pack some "comfort food" as part of your planned food, that's good as well. Something as simple as the juice boxes they are used to drinking instead of just bottled water might make a difference.
Make sure they have good shoes and good clothes. Dress in layers so they can take off layers in the heat and put them back on when it gets cold. Hats are a must as well.
It's going to be even more important to have a plan and a destination if you are evacuating with your kids. In the end, they just aren't going to have the stamina to go as far as you would alone, and you are your wife aren't going to be able to travel as long, or as far, with the kids as you would alone. Figure that into your planning.
With that in mind, I'd error on the side of bugging out "too soon" as oppossed to "too late." I'm not sure where you live, but if you are in a hurricane area, etc, where the possibility of a evacuation is very real, I'd make it a point to keep informed enough so you could decide to evacuate well before everyone else gets the idea and well before any official announcement. The downside is that youre more likely to evacuate some time when it turns out you really wouldn't have needed to leave. If that happens, just consider it to be a "dry run" and see what you can learn from it from next time.
The dog is it's own issue. Don't waste space in your packs with dog food, but put some in the vehicle if your planning to take the vehicle. If you bring the dog, make sure you have a good leash. If the dog can wear a small pack with water, that would help. Don't forget a small dish or bowl for the dog to drink from.
R.I.P HK33K - Gone, but not forgetten.