You should still be able to push start a newer manual transmission car. Put in the clutch, put the key to run, put it in 1st (or so), get it up to speed, let the clutch out, and now the engine is turning, which should turn the alternator/generator which should now be able to run the engine. If your electrical system is really bad, maybe the alternator can't get the voltage high enough to run everything; if your car is very modern maybe the engine control computer won't start up and control the engine before the engine stalls out because of lack of fuel and spark (or the fuel pump doesn't develop enough pressure in time); or maybe the computer just won't do it.

In a traditional automatic with a hydraulic torque converter between the engine and the gearing, you've got a problem: most transmissions use hydraulic pressure to actuate the gear selection, and hydraulic pressure is typically developed by turning of the input shaft. Some older automatics had a secondary pump to develop hydraulic pressure from turning of the output shaft. In those cars, you could select first gear, turn the ignition to run, and if you got it moving fast enough, it would develop pressure, actuate first gear, and then the transmission could turn the engine and off you were. Some references suggest pushing in neutral and selecting first when ready to start. References say you need to get up to about 15-25 mph for that; my VW Vanagon which shares the same engine type as the 914 (and is therefore a rear-engine sports car) can start the engine from a much slower roll; the speedometer rests at 10 mph, so who knows how fast I'm going, but probably walking speed.

As of 2013, manual cars (at least Mazdas) can still be roll-started, as long as the engine computer has enough power to function.

My CX-5 even has a wireless-pushbutton start, not a physical-key-in-the-ignition start, but I've still been able to roll-start it when the battery is too dead to crank the starter motor but still has enough juice for the electronics (lowest I've seen is ~8v if I recall correctly, but don't quote me on that).

The process is pretty much the same: put the car's ignition into the "ON" position (in my case, press the pushbutton twice without touching the pedals -- once to ACC mode, then once to move from ACC to ON), then it's the same as normal: clutch-in, shift to your preferred gear, get rolling, and pop the clutch. Engine computer sees "oh, looks like the engine's spinning, let's add gas and spark" and you're good to go.

Anecdotally, I've seen the described behavior of the engine computer ("detects spinning and adds gas/spark, even if the initial motion wasn't from the starter motor") on automatic transmission vehicles, too. On a 2008 Chevrolet, I found that if you revved the engine up a bit (for inertia), turned the key to OFF, then quickly turned the key back to ON (without turning all the way to START), the engine computer will catch it and keep it running.

I was really surprised when I couldn't push start my 1992 Miata. I had the thing rolling down a hill at like 15mph in first for at least 2 blocks, engine was spinning, but just refused to fire. Jump pack fired it right up. I know the battery was dead after I left the light on, but I figured for sure the alternator would make enough juice to fire up the injectors and ignition...

If you can't push-start a car, it's because it has electronic fuel injection. If the battery is stone dead, there's no juice to run the FI and fuel pump, it will never start. It would work on stone cold carbureted cars because there'd be enough fuel left in the float bowls to bootstrap the whole operation.

No. The clutch must be in when you start to roll the car--the car won't budge otherwise. You get it rolling, turn the ignition to on, then let out the clutch.

I suppose that a 1980s Corolla was the last car I drift-started, though.