Edit in response to replies: Going on vacation with kids is not the same as traveling the world and doing that with the family only works if you're upper class and have no worries about mortgage payments or job security so it's definitely not an option for most people.
Arriving in India with a six-week-old child after an overnight flight taught me something about travelling with children: Problems evaporate.
Everyone was queuing for passport control — our whole flight and several others, probably close to a thousand people queuing in a very large room. One of the minders saw me carrying the baby and waved us out of the queue at once. Would we please go over to the VIP&Diplomat queue. So we did, and another minder waved us out of the queue again. No, we should not queue with the VIPs&Diplomats, we should go straight past that queue and be processed immediately. The whole thing took about a minute.
That was how the trip started and that was how the rest was, too. 10/10 would do again.
If having no problems in a place like this: http://yris.yira.org/essays/1150 is the exception, then what's the rule?
Way too often we think about kids as limiting factor. We try to tell them what they should do, while all we need to do is to show them.
Is it easy? Absolutely not. It's difficult and at times frustrating, but absolutely worth it.
I’m Australian, married with three young kids. Earlier this year, we flew to the US, bought an old bus, renovated it cheaply in Walmart carparks and drove across the continent twice. 20+ states over three months, including a lunch in Mexico. Slept in the bus, in motels, etc. Kids are 1, 4 and 6. It was brilliant.
3 people can live like kings in most of the western hemisphere, thailand, etc. on $20K a year, or live decently on $10K a year. you won't be moving locations often since that tends to be more expensive than just "living" but you can definitely do a stint abroad on a few thousand.
if you need to work, get a remote setup either via contract or remote company, understanding you'll probably need to work a lot less given your necessary expenses.