When a new animal arrives in the home, the outcome rests far less on luck than most people assume, and far more on the skills your dog can already reach for when something unfamiliar turns up in their world, because a dog who has a reliable response to their name, who understands how to disengage from something interesting, and who can sit with a little frustration without acting on it is a dog with a full toolkit, and that toolkit is what carries them calmly through change.
This is exactly why engagement and disengagement matter so much, since teaching your dog to look at something novel and then look back to you, over and over until it becomes automatic, gives them a genuine choice in the moment rather than leaving them stuck staring at a trigger they cannot process, and it is the single most useful thing you can practise before any introduction, because a dog who can take something in and then settle will always cope better than one who fixates.
It helps enormously to read the emotion correctly too, because not every dog who struggles is worried or unhappy, and a dog who seems to find a new arrival hard can very often be feeling pure frustration rather than fear, a desperate wish to go and say hello, and once you understand that you can target the real problem instead of an imagined one, since getting the feeling right changes everything about how you respond and how quickly things improve.
Environment does a huge amount of the work as well, so give the new animal height and safe spaces where they can watch from a position of control, keep the early sessions short and calm, and resist the urge to force any interaction, because animals allowed to communicate on their own terms will often sort the boundaries out between themselves far more efficiently than we ever could, and a few clear signals understood and respected early on tend to settle things for good.
The piece people most underestimate is consistency of presence, because a new pet who keeps appearing and then vanishing stays permanently novel, and novelty is precisely what sets dogs off, whereas the moment that animal simply becomes a constant, unremarkable part of the room the dogs stop reacting and start living alongside them, which is why a puppy raised in the home from day one rarely provokes the same fuss as a pet who comes and goes.
None of this promises an overnight result, and different dogs and different breeds will move at their own pace, yet the principles hold whichever way round you are introducing your animals, so build the skills first, manage the space thoughtfully, read the feelings honestly, and let your pets communicate, and you give everyone the very best chance of living well together.