It depends on the cat. My old cat Leonard seemed to just accept other cats coming in and out of the place without a fuss. He was one chilled out feline... (RIP). :-(
Cats are solitary animals by nature and moving from next door has broken their routine, and also upset the status quo with your own cats, hence the problems. When introducing new cats to a place, it's better to do it gradually, so they all get used to each others scent before being thrown together in one house.
I bet the dogs wonder what all the fuss is about?
