Warning - long post. I'm sure it's too long:
I have two, nearly 8 & 11 y/o. I'm no expert at this but I spend a lot of time thinking about it since my hubby is gone much of the time so I feel like it's all on me.
I prep for anything but our most likely scenarios would be
(1) regional wildfires that require evac to a hotel or to family,
(2) a weather event that leaves us bugging in yet without services.
(3) other, less likely scenarios (EMP nonsense or Chinese hacking our grid down.
BOB Purpose
The kids' BOBs are set up as a combo of "life is normal but we are at a hotel" AND "SHTF and we're possibly walking." Their BOBs are big and they would need help carrying them if walking but I'd rather start out with too much than too little. I know the kids' BOBs aren't tactical but I was aiming in their case for as much normalcy as possible.
BOB Contents
They have clean but gently used clothing from the thrift shop in larger sizes (a couple of t-shirts each, some knit pants, socks, underwear). Also some tennis shoes in one or two sizes up from where they are now. I watch for clearance shoes at Walmart or thrift shops. Just got little one some tennis shoes off of the $1 cart at WM that are a few sizes too big so those go in there.
They each have a book, some crayons/map colors, and a spiral. I will eventually put travel checkers in one and a deck of cards in the other. Each probably still has a stuffed animal since I made them two years ago. They each have a lovey pillow that is small but would be comforting. Each has a baby blanket to serve as extra warmth and comfort even though they are too big now. Each also has a backup flashlight, water purification, whistle, and I will be adding a first aid kit to each.
Family Med Kit
I set up our regular travel bag as a fully redundant first aid kit/medicine cabinet/personal care bag for all four family members. It's very big (backpack sized) but would be grabbed in a bug out. We also use it on every trip so it stays current and I am constantly tweaking what's in it.
Kid-specifically, that bag holds:
liquid Benadryl for reactions/stings/bites, liquid Tylenol for fevers, and Sambucus for antiviral/antiflu. Also a thermometer and spare battery. When they were younger it included saline nasal spray and a nasal syringe before they were old enough to blow their noses.
Mindset
I try to allow the kids to have their childhood and not lay my worries on them too much, but we talk lightly about current events (recent news of Chinese hacking for example). They are very into prepping along with me but think that I do it entirely because of power outages due to weather.
Food Storage
We recently got into couponing and they love building up a food stock. I've started putting it in terms of survival food for them recently. We got 40 jars of our regular peanut butter on clearance last week at Walmart and I said "Hey - we could live on that for a month if we had to! Peanut butter is the perfect survival food." I have put them in charge of organizing our "stock" as we call it and they love putting things away and seeing how much we have built up. I think all kids are secretly hoarders and so they appreciate the value of 40 jars of peanut butter just as a sort of collection.

:
I have two celiac kids. We have to be VERY careful about food storage. I can't stock any wheat products. It is a constant concern with me that we would not be able to rely on emergency handouts or even last-minute stocking up, or eating at a restaurant during an evac, or having family members provide food. I am working hard to build up our food stocks and I keep an emergency food box with 3 days of gluten-free food at the ready.
On pickiness & kid food:
I have one who was very picky because the celiac disease had wrecked her eating patterns and caused vitamin deficiencies that made the pickiness worse (zinc deficiency can cause loss of taste, for example). She's recovering and getting less picky but that scares me still. I have started making her taste foods that she normally rejects even though it causes fights, simply because we won't be able to survive with a picky eater long-term. It's a gradual improvement but I won't let up until she will eat most of what we stock.
My kids actually eat dye-free, pretty preservative-free, as clean and organic as possible, but I balance SHTF concerns by stocking some less-than-ideal foods. In terms of kid-food, I stock hard candy, chocolate chips, sugar

, dried fruit and nuts. When I find anything that is gluten-free kid food on sale, I stock up on it.
Because we can no longer eat out due to the celiac issues, we carry with us at all times water, Lara Bars, bananas, and trail mix. I am so pleased to see that they are learning to just fuel up on those things when we are out and I look at that as a big step towards a better survival attitude.
We just went away for the weekend to a hotel, and took every bit of our food for the three days. That may not sound like a big deal but it felt like a success. We had a Yeti and a microwave and survived on salami sandwiches, whole fruit, PB & J, and then the trail mix, chips, etc.
Skillsets
I'm still a novice gardener but I'm trying to transfer those skills. This year I hope to implement a weekly cook-out night where I learn to use the Dutch oven or we use the grill or other cook outdoors means. Basically I hope to teach them as much as I can before they leave home, like canning, sewing, first aid, etc.
Self-Defense
This is my current sore spot. We have guns and I don't know how to use them so this is the year I embrace guns as home defense. While I"m waiting for that to get going, I plan on buying the kids and myself slingshots and also doing archery practice - we all have bows. They think these things are fun and don't relate it to defending themselves so it's the perfect activity in my mind - non-threatening, but very useful in SHTF situation.
When finances permit, I would love for them to take a karate/self-defense class that is offered here in town. I took it in college and LOOOOOVED it. I need a refresher too.
In summary, I try to involve my kids in all of it while framing all of the activities as non-threateningly as I can while they are this young. As they get a little older I will allow more reality to slip in. I just have to be careful or I will turn into Linda Hamilton from T2 and be raising little John Connors and I'm trying to be more balanced than that.