I have never been in that situation where I had pet birds in the process of raising a family and having to take them on a trip and placing them on unfamiliar surroundings.
I find this very risky, unless your pair doesn't scare/stress out easily and is used to travel by car and you make sure they are perfectly comfortable for the ride and try to make the trip as less turbulent (as in not hitting any bumps, nor any sudden hitting on the brakes) as possible, then the female may continue to stay on the nest and do her job.
Even for some budgies, the fact that we need to take them on a longish car ride for a vet appointment is reason enough for them to be stressed out. If you decide to go through with the move, then you should be prepared for the eventuality of chicks/eggs abandonment and be ready to raise the chicks yourself.
In my case I only had to move my breeding birds from the bird room and into my bedroom and since they all know my house quite well they hardly noticed the difference and weren't bothered by the move. The females wouldn't even leave the nest during the whole thing. But this is of course very different from actually moving away and having the breeding pair to go through travel.
If you have someone who can look after your pair and their eggs/chicks, then that would be the best and safest option. It would be sad for you because you wouldn't be able to follow closely the chicks growth. When the chicks were old enough you could then have the whole budgie family back with you.
Good luck with your decision and congrats on your new job.