In the official Novelisation by Dean Devlin, it's explicitly confirmed that the mothership is somehow providing remote (wireless) power to the slaved ships, including the fighters and city-ships.
“If we can plant the virus in the mother ship, it would then be sent
down into the city destroyers and the attack ships like this one.
Okun told us that this ship’s power was coming directly from the
mother ship, so that must be true of the large ships, as well.”
Once they've successfully destroyed the mothership, the "stingrays" simply drop to the ground.
They’d stayed in the air dogfighting with the last remaining attackers
until the gray stingrays had started losing power and falling out of
the sky. Apparently, they had only limited reserves of onboard energy.
This also explains why the ship in New Mexico was found in such pristine condition:
Like hundreds of flights before and after it, the ship had conducted
several hours of research and observation. It was only moments away
from completing its mission when the parent ship was suddenly
threatened with discovery and bolted away.
The scout ship had wandered further than it should have and now lay
behind the curve of the earth, preventing the energy flowing from the
parent craft from reaching its engines.
