A lot here depends on your definition of "long term". If, by that, you mean a week or two, then you probably won't encounter too many problems, though supplies of disposable batteries in the immediate area will probably become exhausted within the first 48 to 96 hours. Those with rechargeable units, and a viable means of recharging, can expect to have operational equipment for much longer, possibly even years, depending upon the recharging method. Solar, wind, hydro and good old hand crank power sources should be good for several years, and possibly as long as a few decades.
While those sources can supply power for a reasonably long time, eventually anything mechanical is going to break or wear out, and need repair or replacement. Sources of replacement parts will become scarce, and once that starts to happen hoarding will make it even worse, and parts sources will disappear very quickly at that point.
Still, with the number of vehicles out there, 12vdc power will be fairly simple to produce. Anyone with basic electrical and mechanical knowledge should be able to rig up a vehicle alternator to a source of motion (wind, hydro or human) and generate. The real problems will set in a few years down the road as storage devices for that power (batteries) begin to wear out. Your basic Ni-Cad pack is good for about 500 charge/discharge cycles, under optimal conditions, and other battery chemistry types can extend that to 1,000 or more cycles. That's speaking of small battery packs. Large batteries, such as SLA, AGM, or plain old lead-acid, are good for thousands more cycles, but even those will eventually "die" and become unusable. While some boast of "10 year" life, in a PAW I doubt many of those batteries will get the proper care and handling to actually reach that mark. My guesstimate would be that within 3 years finding a working plain lead-acid battery would become difficult, and that within 5-7 years most SLA's and AGM's would have bit the dust as well.
When that happens you'll find yourself at the point where the only way your radio will function is with a direct connection to the generating source, making portable or mobile operation very difficult, if not impossible.
And, of course, this doesn't address the issue of the radios themselves wearing out. Solid state units are going to tend to be pretty reliable, but difficult to repair, or find parts for when they do fail. Older tube type rigs might be simpler to repair, but finding a source for replacement tubes is going to be neigh on impossible in a PAW. Look how hard it is now to find parts for some older rigs, with the full benefit of the Internet and global commerce, and then try to imagine how difficult that will be without those advantages. And, if you do find that precious tube you need for your rig, will you be able to afford it? Anyone smart enough to have a cache of such things isn't going to part with them at a bargain price. They will be an irreplaceable resource, and priced accordingly. A single part might cost your tribe or community half of their total resources. Will you be able to afford that kind of expenditure?
Even the idea of going back to the very early days of radio, with hand built spark gap transmitters will be difficult, because even those required a technology base and manufacturing capability that probably won't exist any longer. Make your own rig? Sure, but where are you getting the materials from?
So, in conclusion, I'd expect that radio in a PAW would be viable, but dwindling, for the first 2-5 years, and sharply declining after that with, assuming conditions continue to go downhill awhile before they get any better, a cut-off point of about 7-8 years where the only radios still on the air will be those in very large communities, with carefully protected stocks of replacement parts and materials, which will be used very sparingly and never for just "chit-chat".
NOTE: Due to the rising cost of ammunition, warning shots will no longer be given.
There's always a certain percentage of any population with the conviction that society is a conspiracy to deny them their rights.
The right to be totally ignorant of any useful knowledge seems to be the basic one. - Keith Laumer