Hi there
I would try and starve him for 24 hours, and then feed small amounts of food (small, frequent meals) of something bland, ie chicken and rice, boiled fish, scrambled eggs (with NO milk or salt). Feed this food, small meals for about 2-3 days, then wean him onto his regular diet gradually. Make sure he drinks plenty of water.
Please make sure he does not eat the cat food as this is too high in calories for a dog, and could cause an upset stomach, and obesity (especially kitten food). You could try and feed your kitten in a separate room, or on top of a high surface (table, window sill...) that Dusty can not get to.
If after this he is still vomiting, you need to take him to the vet. Could you not get an appointment before this date?
Let me know if you have any further questions.
Torunn