As someone who has moved MANY times, with and without a wife, with and without 2 kids, to different cities, states, countries, and continents ... I must say that a figure like "$10,000" -- any figure -- strikes me as dangerously absurd.
There are so many variables at play here, that it would be stupid to ballpark based on some general rule of thumb.
Where are you moving? Do you know anyone there? Do you have a job lined up? What is your profession? How well does it pay where you're going? What is your lifestyle? Do you eat out a lot? What is the cost of living? Are you taking furniture? Are you going to take everything with you? Etc. etc. etc.
My advice is to gather as much information as you can on where you're going, and plan to have enough cash to pay for a fully furnished situation for at least 4-6 months, either sublet or AirBnB, until you find your 'permanent' new home. Or decide that you hate it.
Calculate what your monthly budget will be, including food, utilities, rent, car, bills -- everything. Look at what you spend now, and then find out how much each item in your current budget costs at your destination.
Or just throw caution to the wind and wing it with whatever you have at hand. That's what I usually do in the end.