How to rehome a nest with baby birds?

I'm selling my house in the city and have birds that have built a nest in the second story soffits and are now raising babies.
If I build a bird box put it next to the hole and somehow move the nest to the box then cover the hole will this work?
Is it true that if I touch them their mom will not come back to them?
Any trick to moving a nest?
Thanks for any tips!

If you put the nest box right in front of where the original hole is, and make the hole as much like the original as possible, it MIGHT work. But be prepared for failure.

I've had wildlife rescue people come to several demolition and house move jobs to relocate birds and mamals. It's pretty simple with raccons. If the mom is captured, she and the offspring can be transported to a new spot. If the new den that you choose is unsuitable in some way, the mother racoon will relocate.

With birds, they always leave them until the last few days of the job so that the parents can raise them for as long as possible. Then the chicks are moved to a facility where they are raised by people. This takes a lot of volunteer hours. The parent birds don't follow and they have been unsuccessful at capturing parent birds and getting them to accept the situation of being moved.

I had a job on the ocean where I discovered a nest of young otters. I spent the first day working on parts of the house furthest from the nest. By the next morning, they had been moved to a sheltered thicket 100 ft from the house and we all coexisted for the next 3 weeks. Otters are smart. They seemed to know that I meant them no harm since they were often in plain view, but they were quick to move the babies out of a house that was being torn apart.

So I called the wild life rescue closes to me and they told me they are up to their earballs with wildlife and are deeply understaffed and asked if we could work around the nest.
They told me how to gauge how old they were so we could estimate if they'd be flying by the time my house would be on the market.
So up to the second storey of my house Good lord I'm afraid of highest!!!
Good news is their eyes are open! So older then 2 weeks.

So I did all the painting around that area my self so the big bad birdies don't scare the big tuff guys painting. (to be clear we are talking about sparrows not a raptor)
The plan is to leave them alone until the last possible moment and then scoop the nest and bring it to the wild life rescue.

I learned that birds in fact do have a sense of smell however that is not why they abandon their nest if moved or handled.
Just b/c the parent isn't flying around your head doesn't mean they aren't watching you from a nearby tree. They abandon it b/c they assume if a predator is there and poking around the nest. the predator has already eaten their young, or will return to do so later while the parent is there. I expressed amazement at the crappy adaptation only to be told baby birds have a pretty high mortality rate anyway.

Country life will cure my bleeding heart or I'll have a menagerie of wounded animals in no time