Should there be some sort of property tax capture when a nonprofit sells a building that is converted to for profit use?

I've been thinking about this lately and I don't know what to think.

When a nonprofit sells a property that had been untaxed, and it is converted to a for profit use, should their be some sort of "property tax payment" in arrears?

Baring that, could the revenue from the property sale be taxed as unrelated business use?  And maybe it is, I don't know.

I bring this up because of GWU selling the old Fillmore School ("GW Moves to Sell the Corcoran’s Fillmore School," City Paper), which may have even been given to the Corcoran Gallery, which provided art programs in the building.   GWU has received this and other properties via the dissolution of the Corcoran as an independent institution.

646-656 H Street NE.  Photo from the Washington Business Journal.

Plus, my old nemesis Pilgrim Baptist Church, sold a cluster of commercial buildings on H Street NE ("The 600 block of H Street NE: Going, going, gone," Washington Business Journal), which have been demolished and will become apartments.

However, it is possible that those buildings were taxed since they were commercial, although some were used for nonprofit use or were semi-vacant.

Rendering of what will be built in the place of the buildings sold by Pilgrim Baptist Church.

Note that I am agnostic about the loss of the historic buildings, just thinking about the benefit of receiving a tax exemption and then selling the property for for profit use.