Using alligator leads as heat sink for soldering/desoldering PCBs
I think that using anything to dissipate the heat when doing soldering is just some very nasty folklore, best forgotten unless fully understood. 

On surface mount parts, you want to do exactly the opposite. When soldering or desoldering, you ideally want all parts to be at the same temperature: that's why people use preheat plates. When all things are at the same temperature, there's minimal thermal stress on the terminals.

On through-hole parts with just a few terminals and good access, you may find it possible to grab all the pins at once with pliers to sink some heat away. You'll find that in practice you're more likely to mechanically damage your part by doing that, than just quickly desoldering without any heat sinking. For desoldering leaded parts, the important things are:


Have a high-powered iron. 50W would be bare minimum. Do not use fine tips. The wider the tip, the lower the thermal resistance between the heater and the tip, and the closer the tip temperature will be to your setpoint.
Have an iron with temperature control based on feedback close to the tip. Otherwise your tip, in presence of a heat sink, will be cold even though you think the temperature is right.
Ensure that you have good thermal contact with all the pins at the same time. This means that you need to heat up a big blob of solder with your iron, and then roll it over the pins of the device. In less than a second you should be able to pull it out.
If your device has leads far apart, you will need multiple soldering irons, no way around. Yes, you can try and pull leads out one at a time, but it's a hit or miss if you want to preserve the part.
Use flux-cored solder with electronics-compatible flux. That really means a rosin-core flux. I find that water based fluxes have too low boiling point for rework, and no-clean fluxes are irrelevant since you'll have to clean stuff anyway.
If you can avoid it, do not play with pulling the solder out of the holes before you desolder the part. It's easy to damage the PCB and/or the part while doing so. Apply the heat, pull the part out, and only then remove the solder from the holes so that you can replace the part.