Questions to consider:
1. Why does your husband want to leave his current job? Is it a tangible reason, like the company is probably going under, he's underpaid, there's no chance for promotions or growth and as your kids grow older his current salary won't be enough? Or is he simply bored with his job? Does he hate the commute? Does he dislike his boss?
2. Why does he want to move? Is there a guaranteed opportunity, or is it more like "I heard that California is a great place to work" or "I think I could find work in Florida"?
3. You know your kids can't stay in their present schools forever, right? I'm thinking the 10 year old is in middle school, and that kid will have to change schools and go to high school? And why are you allowing kids that young to try to influence their dad on such a momentous decision? If you're at Disneyland, let them beg their dad to take them on the scary roller coaster. But they should NOT be begging him to stay with his present job if he believes that a different job in a different location is better for the family. They don't understand seniority, pension plans, the workplace environment, professional qualifications, promotions, chain of command, etc. To even discuss this with them, and to allow them to beg him not to move, has seriously undermined your husband's responsibilities to his family.
If your husband has found a good job in another part of the country, where he knows he will be hired, where he will find satisfaction, please support him. If he's thinking of leaving his job as a teacher or IT consultant or plumber or a factory worker or surgeon, and hoping to become a tattoo artist on a boardwalk, or dreaming about opening a snow cone machine kiosk near a beach somewhere, when he has no artistic inclinations and has never operated a food establishment, then please encourage him to talk to someone about how to resolve his dissatisfaction with his current employment and living situation. Maybe his parents' moving away stirred something in him. Maybe he was only living there in hopes of helping them out. Now they've moved, and he's free to do something he sincerely wants to do.
We've moved a lot, to different countries, continents, states and even an island, due to my dh's job. We told the kids they would have different friends, different houses, different rooms, different foods, etc, but they would have the same family, the same traditions, the same things that are important to us as a family. When my son graduated from high school, he had attended 10 different school systems. Some he loved, some he tolerated, but it created in him a sense of independence, resourcefulness, creativity, and adaptability.
Anchor your family with love and security, not with a particular zip code or street address.