Every household with children ends up with a bag that gets grabbed on the way out the door. The trick is keeping it useful after the first month. Two houses double the work, and the bag at Mom's place is the one that gets forgotten.
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Decide what goes in it
Keep it small enough to carry in one hand. A change of clothes for each kid, sized for this season and the next size up if you can spare the room, a pair of socks, a snack that keeps well, a water bottle, a phone charger that matches the kid's phone, and a copy of the emergency sheet from the fridge. Medicine does not go in the bag unless the family, the doctor, and the pharmacist have already set that up. The bag is for clothes, food, and paper.
Pick the owner
Write one name on a sticky note inside the bag. That is the person who checks it. It can be you, your partner, or the adult at Mom's house, but it cannot be 'everyone,' because everyone means nobody. Two bags, two names, written down.
Set the rotation
Pick a day, such as the first day of each split week, and run the same short list: check the clothes against the kids' current sizes, toss anything past its date, swap the snacks, and make sure the charger still works. Put the rotation in the shared calendar so it lives in the same place as the school events and the pickups.
Plan for the bag to go missing
Kids take things. Bags get left in the back seat. Keep a spare set of the basics in the other house's bag so a missing item at one door does not turn into a dead Tuesday at the other. Ask the kids to help choose the snacks, because a snack they picked will get eaten and a snack they did not pick will end up in the back of the car.