I understand that much of it was rendered inoperable by the retreating BEF. But did the Germans end up using any of it, or was it incompatible with their forces? Was it melted down and recycled? Did the French Resistance make use of some of the equipment and ammunition?
I doubt the French Resistance would have had the time to sneak in to steal disabled equipment.
And generally the Germans would not have made much use of anything from Dunkirk, though the Germans in their wider campaign in France did re-purpose some vehicles. Hotchkiss (H35) light tanks, and B1 heavy tanks were adopted by the Germans for mostly occupation minded uses. The wider issue, though, was simply that there was a lack of real compatibility between French, German and British equipment.
Putting it in perspective, the French and British had entirely different ideas from the Germans as to what made a good tank. Where the Germans largely stressed fast, highly mobile armored divisions, the French and British were still in the mindset that tanks either functioned as highly mobile "cavalry" tanks (See British "cruiser" tanks, and French AMR 33) and thicker armored tanks intended to support infantry (see British Churchill Heavy Tank, French H35, B1 heavy) where as just about every German tank in this period was purely about mobility. The Panzer 2 was basically just an armored car with treads instead of wheels, and a high caliber machine gun. The Panzer 3 was a bit better in terms of armament, but was also intended to fight armor.
Furthermore weaponry lacked common calibers. From machine guns to rifles and side arms, there simply wasn't much of anything that could be picked up and immediately incorporated into the German army, though, again, much like with re-purposed tanks, there is, at some level, domestic arms being used in occupied sectors.
If the Germans made use of looted, and acquired equipment, it was out of desperation and immediate need. There was no realized effort to mobilize France's industry to supply Germans with tanks in the war. Or guns. Or equipment. Ergo, anything that the Germans seized at Dunkirk would have likely either found a minor role (target practice, training vehicle, demonstration, left for occupying forces) or scrapped for parts and raw material.
I can't add much to what crassbar said but just as an interesting side note/back up of the facts proved, my grandfather was captured at St Valery during the evacuation as part of the 51st Highland Division. When asked to surrender by a German in a tank he threw away the pin from his gun and jammed the barrel into the dirt to damage it, chap in the tank just laughed as they had no need for British weapons.