In order to ensure a smooth session, there IS some prep work you can do. First, explain to your kids what's happening - you're going to a cool spot to take a few pictures and have a little adventure while someone follows you around with a camera. Second (since you're the one reading this), explain to your family how important this is to you - to have pictures of them now so you can remember this time when they're older.
Third, PACK SUGAR FOR DURING SESSION BRIBES. My favorites are pictured below - they're small, easy to chew, and don't stain clothing or mouths (looking at you, blue m&ms). Fruit snacks, smarties, marshmallows, necco wafers, anything little, light-colored, sugary, and quick to eat is a win. If your children have food allergies, make sure to alert your photographer in the event they bring their own bribes.
Fourth, make sure your kids are rested. Most photographers (myself included) prefer golden hour for outdoor family sessions. In the late fall and winter, depending on where you live, golden hour is early...maybe even before dinnertime. But summertime? It's laaaaaate. Your kids might need a nap or rest during the day. So maybe skip the pool or park that day to ensure well-rested kids who will be put straight to bed after their post-photo-session-tick-check. Lastly, and this is a during the session reminder, kids are kids. Your photographer isn't judging your parenting, or your kids, or you. They're just trying to get beautiful photos of your family, which in the course of a photo session will have ups and downs. The happier you are to be there in that moment, the more that happiness will show in your photos.