As far as I can tell, the western allies didn't have highly mobile armored or mechanized formations above the division level. In contrast, the Germans had large armored formations called Panzergruppe and Panzer Armies, lead by such noted commanders like Heinz Guderian and Edwald von Kleist. The Soviets, too, had Tank Corps and Armies which spearheaded their counter-offensives at Stalingrad and Kursk. Was this a reflection of different doctrines in use by the belligerents or is there a different reason?
I'm waiting on a class to start but some considerations to take into account are things like:
Panzergruppe and Panzerarmee are that in name only. Look at their actual composition. Though based on a core of armored divisions you'll find a majority of it to be regular infantry divisions with a small amount of panzer grenadier divisions to help exploit what gains the panzers would make. Especially in the late war when they would call many groups these names but at best they were a very wore down Corps.
A Soviet Tank Corps in 1942 and later was a completely different organization than the Allied or German militaries. Comprising of two tank brigades and an infantry brigade that had a total armor count that would equal out to their opponent's tank division in strength.
I would say to concentrate on efforts in the organization and usage of armored divisions as that is probably the largest yet easiest maneuvered formation. When you get into scales of armies and army groups and fronts we're not looking at a lot of general movement than an inexorable drive forwards or backwards. Also of note are the independent heavy tank battalions of Germany, the independent tank and tank destroyer units of the US and such that offered a bit more armored flexibility to units not organically equipped with armor.
The German term Panzerarmee/corps was given to formations with a higher number of motorized/tank units that doesn't mean those formation were more mobile than their Allied counterparts. They just were more specialized and motorized than other German units. The Allies just didn't specifially name their corps like that. The corps were numbered regardless of composition.
Normandy as example:
During Operation Goodwood the VIII corps consisted of 3 Armoured divisions with about 750 tanks. The 1st SS Panzer Corps had around 200. It had the name "Panzer" corps because it included the Panzerdivisions 1st SS and 12th SS but this doesn't allow for any conclusions regarding their mobility or equipment i have seen Panzercorps without Panzerdivisions actually.
The Western Allies just didn't use such terminology.
The average Allied infantry division was often better equiped than a German tank unit even considering tanks. During the cause of the campaign Allied infantry divisions often fielded more tanks than the opposing German Panzerdivisions.
While the doctrines differed the naming of formations has nothing to do with it.