New puppies are awesome. Seeing a puppy causes the same physical reaction that occurs when seeing a baby. It gives us the warm fuzzies and causes the release of oxytocin, the hormone that causes bonding and brings out our protective instincts.
Puppies don’t know the rules, tho. Puppies, like humans, learn thru play. They learn what causes the play to continue (polite behaviors) and what causes the play to stop (impolite behaviors). Play may be with other dogs but it should also be with their human. One study found that proper daily play time reduced the likelihood of up to 22 different behavior challenges.
When bringing in a new puppy to your family they need to learn the rules. An older dog may give them a free pass on play behaviors that may be unwanted or too over the top. They may correct the puppy with a grab of the snout or they may turn away or walk away. At some point, tho, you may need to intervene.
Puppies are like toddlers in that when they are tired or hungry those over the top behaviors start to happen. Puppy piranha mode (where they want to mouth everything and anything they can) will often happen at this time. Offering them something to chew on aside from your other dog will be a big help.
Watch for signs of stress in your older dog (licking, ears back, whale eyes, constant shifting away from the puppy). These stress signs signal that it is time to intervene and either engage your puppy in a play session with you, such as tug of war or throwing the ball, that allows your older dog time to relax and regroup. Provide both your puppy and your older dog with a self entertaining activity, such as time on a snuffle mat, stuffed kong or a new chew toy. This reward for an older dog will help reinforce that a puppy can be a good thing and will remember that reward instead of just the piranha puppy.