A likely cause is because the pellets of the dry cat food are too small. You cat swallows them whole in quantities sufficient to make him nauseous. He is ingesting unmasticated (unchewed), hard objects which sit in his stomach. It is not surprising that he might throw them up. This is one, good potential reason which needs to be checked out.
No one else is mentioning it not even vets as far as I know and this problem has been around for years. I have written about this before years ago and I think that the pet food manufacturers came up with the idea of large pellets from my observation but I am probably suffering from an inflated ego on this occasion. Nowadays cat foods made with large pellets are made for oral health. They force cats to chew them which helps to clean their teeth and gums. That’s the theory. I’d suggest buying “oral health” dry cat food from Hills or Royal Canin in a small bag initially and try that. If it stops him vomiting buy one of the large ones (5 kg). Or better still give him wet food.
CAVEAT: there is a big caveat to all this. You have to be completely reassured that your cat is not vomiting for some other reason. This box must be ticked. Only a vet can do that with the highest degree of certainty. Vomiting is a generic symptom. There are many causes. And cats vomit well naturally. These factors can make it tougher to find the underlying cause.
You may be able to avoid a vet by putting your cat on wet food or large pellet dry food and see if he stops vomiting. If so, job done basically. You would have found the cause: dry kibble made with pellets that are too small.
Which brings me to the question why? Why do pet food manufacturers make dry cat food with pellets that some cats will swallow whole? You’d think that the manufacturers would firmly eliminate that possibility because it is bound to cause problems. It shows them up for careless, lazy businesses, which they are. It is still available.
In the 1970s and 1980s (the beginning of the popularity of dry cat food), when dry cat food started to result in a very large number of cases of cystitis (bladder infections) in cats, the manufacturers cynically manufactured more dry cat food as a cure. It was the same stuff but a different formula (more acidic and low in magnesium). The real cure should have been wet cat food. In my opinion, they avoided it because they wanted to push dry as it was going to prove more popular and I presume made them more money. Or they were just dumb.
P.S. Cats have a low thirst drive so don’t fully make up for the lack of moisture in dry kibble. Cats would normally get a lot of their water from their prey which is made of 70% to 80% moisture.