Dogs can get digestive upset from abrupt change in food, particularly kibble. The main question is whether that’s what did it or whether it’s a coincidence. It is likely that’s what did it, though. That’s why gradual change is recommended.
Another thing to consider whether your dog is dehydrated at this point, or in pain, or in more serious distress than the diarrhea itself. Still acting normally? Playful? No lethargy? I any additional signs of concern, I’d contact a vet.
If not, I’d let the system rest for a day with providing just water or potentially home made broth or bone broth. Then the question is whether bland diet (such as boiled chicken with rice or white potato) is a better idea or whether it’s best to return to the original kibble, let things settle down and if you still want to change do it very gradually.
I’d take him to the vet as soon as possible. A cough can indicate many things other than kennel cough or something stuck in the throat. It can be an upper respritory infection, the start of a heart problem…some scary things. The best thing is to see a vet.