Why Neville would run downstairs from a shooting Jeremy:

[Adam]

To get to a phone. The upstairs phones had been disabled by Jeremy upon entrance to WHF.

To leave WHF and get to the Folkes's.

June had been shot in bed, probably fatally. She would be of no assistance.

The twins may already be dead.

With or without bullets, Jeremy was dangerous. He was a fit young man, fully clothed on a mission. Neville needed a weapon or to escape.

Neville followed Jeremy when he went downstairs to reload. To try to prevent him returning upstairs fully loaded.

To get the bullets which Jeremy would need to re load.

It was a natural reaction to run. At first not knowing who was shooting him and June in the dark.

Neville would not know how many bullets the marksman had. He would assume the marksman knew about guns and would be looking for a head shot.

In the dark Neville would not know what gun the marksman had. It could be a powerful one.

[lookout]

 Cobblers.

[Adam]

The last two days have been very productive.

There is no good reason for Neville to abandon his family.

There is no reason why June would still be in bed with her head on the pillow when getting her first shot.

There are lots of reasons why Neville would go downstairs if Jeremy was the shooter.

[scipio_usmc]

If Nevill didn't know the gun was empty he could have fled to avoid being shot more.

Nevill also could have ran to the kitchen to get a weapon.  The guns and ammo were in the back kitchen closet.  The closet itself had knives and other potential weapons.

There is no way to know whether his killer ran to the kitchen and he gave chase or he ran to the kitchen with his killer chasing him.