Could be a dodgy ground...
If the head unit's turning off, it does sound like poor current flow for the head unit, or bad ground on an amp.
The latter issue can act up very badly if the amp is able to see 'ground' back through the RCA cables to the HU!

I've seen this, and it's ugly! The entire amp current load is going via all the worst places.
Try refitting each component's -ve connection to the chassis. It needs to be a good, solid, paint-free link.
If this persists, I would try running a temporary +ve cable (a good thick one) direct from the battery to each component. Then turn it up again, and see if this resolves the problem.
That's the cheap & easy things to check first, anyway.
- Rich