If you are determining whether a dog will accept a new household with existing dogs and vice versa, consider trying the following steps. While these are not exhaustive tips on integrating dogs, they really helped with Ginger when my parents took her. Please consult dog behavior books for further instruction or consult with dog professionals in your area for assistance if there is any doubt. Special care should always be applied so that none of the humans, dogs or other pets are injured.
1). Let them meet in neutral territory - take them for a walk.
Let the dogs meet for the first time near the final home but where neither sees it as their "territory." Have the current masters walk each dog, and then start walking side-by-side toward the final destination (the new home, or even the new owner's car, if in different cities). If there are multiple dogs in the new home, a separate person should walk each dog.
The goal is to walk them until they are all just really tired and end up at the new home/new car. Let them sniff briefly initially, greet, and then repeatedly for brief periods along the walk, but keep them moving on the walk - which is their common task.
Let the environment be casual as possible, "non-eventful" since that will tell the dogs they can be relaxed too and enjoy the walk. As they start sniffing stuff together they might begin seeings themselves as a pack - and dogs are naturally pack animals. If they get too assertive with each other on any of the greetings on the walk, pull them apart for a few feet, keep walking forward, and then try to see how they later walk together. Let them greet again later, each time briefly and casually, with each master having strong control over their own dog.
Any inappropriate actions can be met with a strong, "No!", but then forgotten and keep them walking. This will help the dogs develop familiarity with each other, and allow the masters to determine if the dogs can ultimately live together. If the walk goes well, then the end result is that the dogs end up at their new home/car. All the dogs should be "dog tired" at the end and then give the dogs all water. They most likely will simply lay by each other and that will show some how they will interact - and if they will accept each other.
2). At the new home or the new owner's car, let it be casual with the existing owners/"parents" so that the dogs still get signals that everything is causal and fine.
If everyone is at the new home, have the old owner come in as well, so their dog doesn't understand at this point there is a shift in ownership and will feel more comfortable with the home. During this process, the owners may at times shift leases so they are holding the other's dog and that will also tend to convey to the dogs that everyone is friendly and there is no need for concern.
If these efforts don't allow the dogs to be relaxed and there is any consistent, high level of aggression between them - particularly when they are tired, then it may show that the effort may not work - or needs professional efforts to try to start the integration of the dogs. Each person holding a dog's lease should be comfortable exercising dominance over the dogs so that the true "alphas" are understood to be the people.
3). The Dirty clothes 'trick' can help with a crate.
Once the old owner leaves their dog, one idea that can help their dog transition into its new home is to also give a bag of dirty t-shirts/towels that belong to the old owners and have not been washed. They will have the scent of the old owner on them and can help comfort a dog if the pet seems sad/lonely when the old owner is not around. It can often be helpful for a crate also to be used as a place for the new dog to sleep, so he/she may sleep apart from the other dogs. Ideally, the crate door should be left open initially so the dog does not develop a negative impression of the crate. The old master's dirty clothes can be put in the crate as bedding, and allow the dog to feel safer as they adjust to their new home.
If the crate with old clothes is used, the new dog may be territorial of it if the other dogs enter the crate since the old master's scent is in it. In such instances, take care when other dogs are allowed in the area. The crate might be placed in a separate room, with the crate door left open for a long while. Once the dog goes in there, and the dog often will feel safe, so the door can be shut later on. In time, the clothes can be taken out in a day or two so that the dog starts bonding with the new family.
4). Separate at feeding/sleeping/alone times.
Food aggression is one of the most common reasons for flare-ups with a new dog and the existing pack. The dogs should be separated at feeding time until the pack's roles are clearly defined and accepted. Dogs should also be separated in a similar fashion when they sleep or are alone without supervision.
5). Special care with particularly smaller pets and cats.
With a new dog in a home, special care should be taken to determine if prey aggression will impact birds, cats, and other smaller animals, even small dogs. With such size differentials, it can be dangerous since the smaller animals will not be able to fend off an assault. If the new dog attempts to show any aggression with the smaller "prey," then likely the owners need to understand that this may never stop. For at least small dogs that are sufficiently aggressive themselves (many terriers), the danger may be less since they may be able to protect themselves.
However, with cats specifically, they may not be safe with any prey aggression. If the cat is declawed, then the cat must be protected at all times. Understand that the cat's only remaining defense of biting is INSUFFICIENT since cat jaws are very delicate and that the cat is at serious risk of a jaw fracture if they have to defend themselves against a dog.
Again, these are not exhaustive tips, but can be helpful.