Only if they have enough income to offset the expenses.

So if they never have income for that business again, they will never be able to use it?

It does not get triggered if the close the business or sell the property like for a rental business operation?

So if they never have income for that business again, they will never be able to use it?

Correct 

If I remember correctly, it stays with the house, not the business.  So if a different business with a home office in the same house is started, the carried over expenses can be used for the new business.

But as you said, if/when the house is sold, it is forever lost.