I've never ran out but did just get 2 wheels in my garage(where the EVSE was) once when I was trying to run the car very low to do a 0-100% charge. I was able to keep the car on and in neutral and with the help of a neighbor, push the car in the garage to charge it.
My daughter got even closer to total discharge, she only made it 5' short of the garage and the car totally died. Wouldn't even go into neutral to push it the final 15'. As my garage EVSE wouldn't reach I had to resort to the 120v EVSE and an extension cord. After a couple hours on 120v I tried starting the car to get it in the garage and plugged into my L2 EVSE, but it was still too low to go into gear
I was able to shift into neutral though and with help was able to push it into the garage where it sat for the next 6hrs charging on L2.
I really doubt a tow truck would carry around a L3 EVSE and my experience tells me even a L2 EVSE may require several hours to get the vehicle to move very far, no I'm guessing they'd probably just tow it.
This also kind of shots my original idea of carrying around a 12a 120v portable Honda generator, if the Leaf got that low it would literally take hours and hours to do any sort of charging, no you'd need a 30a 240v model and they weigh hundreds of pounds and would fill up the whole back(and back seats) of a Leaf. Best to just plan your trip as to not run out of power or have several backup plans if your charge starts getting low.