Did you see that my steer, ‘Weber’ is STILL grazing? We’ve never had grass so green & long this late in the season. It is terrific, is saving on hay costs for him & Ithaca and he is just loving being moved around the property and chowing down all day long. :)
We’d do the same with the mule, but he’s SO stubborn. You could be dragging him to a field of fresh CLOVER and he’d still fight you every step of the way! Anything and everything needs to be HIS idea, and HIS idea, only. He’s too smart for his own good, sometimes. ;)

Really nice that you still have grass! I always loved it when the horses had grass into early December ... saved about a month of hay and at one time, I had 5 mouths to feed.
My first pony was a smart old gal - we could tie her out in the yard (& did a lot) and she knew how to keep from getting her feet tangled up in the ropes .... the big horses, I didn’t dare put out on a rope!
I was watching a TV show some while back (like maybe 2 years!) and the guy had mules. He left them at a farm during the winter (he was back in the boonies, didn’t have food for them) so in the Spring, he would go get them and take them home. Well, he tried to take them home. One mule started acting up & the other mule seized the opportunity to also act up. He ended up walking back to the corral after being dumped (the mules were ‘waiting’ for him) and he was trying to figure out what to do. His comments were that you need to be exceedingly careful with mules. When training, if they have a “bad” experience, they never forget it. He was in a bit of a pickle because his mules ‘got away’ with misbehaving, which was problematic.
My mom, growing up, had to plow with mules. The one that left a lasting impression was “Bill”. He was a good mule in general, great worker, never stepped on the plants, knew exactly how to turn around at the end of the rows. BUT ... if Bill even ‘thought’ he heard thunder, he was headed back to the barn, regardless of what he was doing, hooked up to, etc. No thunderstorms for Bill!! :-)
You might just try a bouquet of clover to lead him :)