I have a large household -- myself and at present 7 dogs (3 of the puppies of various sizes and ages) and as of today, 16 cats (4 of the kittens from 8 weeks to 15 weeks old). They all come to me from the street as babies or youngsters and are mixed, so it is impossible to know what all their genetic hardwiring urges them to do. Dogs love to chase, and you have to be very, very firm. I let the puppies and kittens socialize with each other and with the older animals in the house for some weeks before I every let a youngster out into the garden with the older animals. In a more closed environment, I can reinforce proper behavior on all sides (the dogs not to bark or chase, the kittens not to scratch, growl, or bite). I also make it a point to have them learn to accept other humans by consciously having them in the same room with visitors (unless the visitors are the silly types who are terrified of getting a scratch or a nip from one of the babies - but then, those kinds usually don't come back a second time...). I discovered that, if I was the only human they knew, they were shy and sometimes anti-social with strangers.
When I started out, I was rather ignorant and, because they loved each other and slept in the same laundry basket at night, I allowed a rapidly growing puppy to play at will with a kitten of the same age. Only, at 6 months the puppy was surpassing German Shephard size (she is a huge, slender, graceful grayhound and retriever kind of mix), while the kitten was still quite small. One day in rough play the kitten made a leap for the window and the dog, not wanting him to leave the game, snatched at his hind legs with his mouth. The cat had a broken pelvic bone and other problems with the head sockets of his thigh bones. Four months in a confined cage, and two operations later, the cat and the dog were still together, but I had gotten some sense and taught the dog proper manners and some important commands. A mistake I did not repeat. Now all socializing is initially done under supervision, and I am very severe about dogs chasing even if, as is often the case, the cat invites the chase.
To end this long story, the numbers of the household inspire female dogs to adopt kittens, and cats to adopt puppies, and both puppies and kittens to adopt each other as siblings. Each new introduction to the household creates some stress and problems, but when the pride and pack are both in harmony, then newcomers are much more quickly and peacefully folded into the family. I have not yet taken in a stray pup who wanted to kill cats, even though many of them are mixed with breeds that normally are dangerous to cats -- terriers, doberman or rottweiler mixes, hunting dogs of all kinds. But then, I never, ever take in an older animal. I do not have kennels, and all of the animals have to share the garden with me and with each other. I spay females at 8 or 9 months, and neuter the males around 6 months. Thus, socially, we are a community of orphans and adoption -- we become a tribe of 3 interlocking species and no longer have much experience of customs or behaviors practiced by outsiders. Most of the cats become more dog-like and most of the dogs more cat-like, and all learn my moods and fancies, as I try to adapt to theirs. So, rarely any chasing or nastiness between species -- only occasional inter-species postureing...
Just be careful with the dogs, and be very, very strict about the essential forbidden behaviors.