The answer is "it depends". Sadly trials usually measure how long an engine lasts in hours, not miles, so it's hard to translate directly, but you can see the relative reliability.

The Sherman used 4 different types of engines (or 5 if you count the M4A6, but it never saw service).

The first is the R-975 radial engine. It came in three different flavours of increasing reliability: R-975-EC2, R-975-C1, R-975-C4, with Shermans chiefly using the C1 and C4 variants that could work on reduced octane gas. Medium Tanks M4 and M4A1 (as well as the M4A5 if you count the Ram as a Sherman) tanks used this engine.

In trials to see if the engine could last 400 hours of running or 4000 miles held in 1943 (whichever came first) the R-975-C1 performed the worst. Not a single one performed for longer than 200 hours and servicing the engine took 132 hours during the trials, which was the second highest (but the engines that had the highest also ran for much longer). The average lifespan was 166 hours. Note that these trials tested only the engines, everything else (including engine accessories) could be replaced without limit. This was better than the EC2 variant (M3 crews complained that the average lifespan was just 100 hours even in training conditions), but still the worst out of all the Shermans.

The modernized R-975-C4 performed better. Trials of three new engines showed that they lasted for 177, 219, and 231 hours respectively, a C1 modernized to C4 standards worked for 222 hours.

The British had a similar experience with the tank. Reports from North Africa stated that the tanks, if maintained very carefully, would give 180-200 hours of service and an overhaul at 100 hours. In the African desert this was 700-900 miles of driving. Later, in Italy, engines were reported to last for 225 hours (no model is given, I assume these are the modernized C4s), but the R-975 never truly caught up with its brothers. There are complaints even in 1944-45 of radial engines only giving a few hours of service. The British 38th Armoured Brigade complained that radial engines give significant drop-off in power after 400 miles of running and 20 had to be replaced by August 3rd, 1944, after under 900 miles of running.

Then we have the GM 6046 twin diesel. In the 400 hours/4000 miles trials one engine reached the 400 hour mark, the rest broke down at 276, 278, and 353 hours. Sadly I don't have combat figures from Africa in hours, but the report gives 1500 miles, so the tanks were quite a bit more reliable than the M4A1 in practice. The M4A2 seemed to suffer a lot more in Italy, units reported a drastic drop in performance after 1000 miles of running. Most of the issues they complain about are track and transmission related though. Tanks with this engine were sent to the Soviets as well, they estimated the lifespan of one tank to be 250-300 hours or 2000-2500 km by the end of the war, although early engines proved very unreliable and would sometimes be put out of action after just a few hours of use.

Third, we have the Ford GAA engine used on M4A3 tanks, the American favourite. In the 400 hour/4000 mile trials Ford engines lasted for 293, 302, 347, and 350 hours. Trials of a tank sent to the British (they got a sample even though they didn't use it in combat) were stopped after 259 hours.

Finally, the strangest engine of all, the Chrysler A57 Multibank used in the M4A4 Sherman. This engine was composed of 5 6-cylinder engines attached together and when it was first shown to users they were afraid it would be terrible to maintain, but the reliability was actually fairly high: in the aforementioned trials one engine died after 339 hours and three survived for the entire 400 hour race.

Now, the major limiting factor in the lifespan of a Sherman in real use wasn't the engine. It was most likely either the tracks, suspension springs, or road wheel tires, which did not last anywhere as long as the engine did. Different types of tracks and wheels gave different results, and springs were improved as the war went on, but I don't have compete data on this (u/The_Chieftain_WG might very well know). From what I can tell, there wasn't a marked difference. The British lumped all Shermans into one heap. Shermans were considered Class I condition up to 1000 miles of running, Class II at 2000 miles. Only the Cromwell and Valentine had similar reliability, the older cruisers were considered Class II after 500 miles, the Churchill (no mark given, but since this standard was set in 1943 then probably the III and IV) at 400, the Matilda at 350.

In conclusion: would a specific Sherman have needed a major overhaul after 500 miles? Quite possibly. Would a unit equipped with Shermans be out of action only due to mechanical failure after making a 500 mile march? Almost certainly not.

References:

https://warspot.ru/8378-glavnye-tanki-yanki

https://warspot.ru/13936-amerikanskiy-general-v-rukah-anglichan

https://warspot.ru/14340-afrikanskiy-debyut-shermanov

https://warspot.ru/15856-sherman-dlya-russkih-i-anglichan

https://warspot.ru/15997-emchi-po-druguyu-storonu-kontinenta

https://warspot.ru/16646-samyy-amerikanskiy-sherman

https://warspot.ru/15434-vundervaffe-po-amerikanski

http://www.tankarchives.ca/2013/06/tank-reliability.html

Canadian Military Headquarters, London : RG 24 C 2 reel C-5772 image 4189-4190

Canadian Military Headquarters, London : RG 24 C 2 reel C-5773 image 5388-5398