So, this is now getting into my head.
It's in the middle of the Two Towers, and Pippin and Merry are being dragged off by orcs through Rohan, followed by Strider, Legolas and Gimli.
The King of the Rohirrim, Théoden, is disabled by magic, and Wormtongue almost rules in his stead, but Wormtongue is weak, and the rest of Théoden's family is not, and Wormtongue Master cares not for him. Not that'll last too long, but it'll slow responses.
And, so, as a horde of Dothraki, men women and children, appear in the lands of Rohan, none are ready.
Khal Drogo is a thug, as much as anything else, but he's not an idiot. Where is he? What's out there? He doesn't know. So, smaller groups to scout out whats there. But, they're going to be Dothraki scouts, and they're effectively almost at war with everybody. Slavers, raiders, and unwilling to respect any outside their own. They're not going to treat any Rohirrim they meet well, unless they are forced to. But, any Rohirrim who's ready to fight, properly equipped, is going to outfight them. And match them on horseback, which is something they're going to blink at.
After that, it gets more complex. Rohirrim taken as slaves are not going to be easy to play with. Wormtongue's games with Théoden are going to run into problems, because who'd respect either of them being weak in the middle of an invasion? Added to this, Khal Drogo's going to find a hard riding, hard fighting people, whom even the women tend to fight back. Do they even have enough food to steal? Does his Dothraki have enough for long? Just basic supplies is going to make things interesting.
Meanwhile, there are others who will react. Saruman has a army, well equipped, trained monsters. And these invaders are screwing with his plans! His monsters will march, but to where? And against whom?
How about the Dunlendings? They're near, and much weaker than the Rohirrim. Will Khal Drogo conquer them first? They're ripe for it, compared to the Rohirrim. And this is yet another chance for Saruman to support his "allies", but does he care?
Then, there's the remains of the Company. Small in number, but no patrol of Dothraki is going to be able to deal with them. They're another wildcard, heading out on their own busness, and smashing much that tries to stop them.
Note, in canon, Khal Drogo was entirely willing to talk, and go with plans he liked. You could negociate with him, and it was possible to have him stick to it. If the Dothraki run into orcs, that's almost certainly going to be a fight. Monsters? Where they roam? Yeah, that'll be a mess. I could honestly see Khal Drogo running into orcs, then Gandalf and Legolas are going to be real proof that there's more going on. Monsters, fallen demi-gods, elves, dwarfs and more. Dark Lords moving, heroic times, and a call to all free people! Will this sway any Dothraki? Likely not many, but the way the Rohirrim got their current lands was a treaty with Gondor. Would Khal Drogo like that idea? And, if they ride into battle next to the Rohirrim, how many Dothraki are going to see the Rohirrim as them, but better?
Yup, heck of a mess. Could make a great story, though.