Ask for a hand when you've got the movers in
The easiest, cheapest and quietest way to take care of kids when you call the movers in is to find a helpful friend or family member. Your little movers will be much happier with someone who can give them plenty of attention for a few days. That is, while you and the movers sort out your new home.
Perhaps a friend from school, family member, or long-time friends can spare a couch. Having a plan for your junior movers will help them feel more secure. They'll be happily anticipating a new home and your movers will certainly thank you.

Find fun holiday activities for junior movers
If you call the movers during school holidays, check on holiday camp options. Even a couple of days can be a help. And your kids will be happy and active while you and the movers get heavy furniture into place. Not as cheap as a benevolent friend, but your head movers will come home tired, happy and ready to settle in.
Get your mini movers to help
Alternatively, incentive programs often help mini movers. Get the kids involved. Perhaps offer a reward for packing toys into a moving box or let them choose the shelf for their books. You can always reorganise little movers' madness later. At least things will be put away. Junior movers usually respond well to special tasks and responsibilities. Like wiping down furniture after the movers have finished in a room. Or making the beds. Every mover knows the joy of falling into clean sheets at the end of a relocation!
Make the whole occasion special for your mini movers. Give them something unusual to do. And make sure you remind them of what a big help they are to you and the movers. Most importantly, try and keep children away from the movers. It's not only practical but important for their safety.