Dogs have their own way of communicating that involves their body language and face, such as a submissive grin. One behavior, called a "play bow" might explain your question about "why my puppy keeps laying down when walking."
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Dog lays down every time
First of all, puppies may lay down when walking simply because they are tired. But beyond that, a dog's play bow is an illustration of a behavior that some dogs will often do when they see other dogs. In this movement, the dog's front paws and body are low to the ground. Cesar Milan explains that the forward movement indicates excitement and the low body is meant to be interpreted as friendly and playful.
An aggressive dog can also lower their front body to the ground, so it can sometimes be hard to know when the movement is threatening or when it is playful. A dog that stops when he sees other dogs may drop to the ground in play or may run away in fear, or may run away to indicate that he wants to play.
Milan says interpreting the difference relies on watching his head, ears, tail, and back. If these are perked up high, it means the dog is feeling dominant. If they are low, they are feeling submissive or uncertain. Tension in the dog's muscles, particularly in the back and legs, can indicate a high energy level that can either turn into playtime or a brawl.
Dog crouches when I approach
Dogs don't just exhibit this play bow behavior with other dogs — they can also do it when humans approach. A dog may come charging at you and put her body on the ground as a way of showing her excitement. Modern Dog Magazine says the play bow posture is usually a basic invitation to play. They say the dog will usually only hold this position for a few seconds before starting to run away, according to Philosophy of Dog,
If your dog stops when she sees other dogs and this is a behavior that bothers you, some training to help her learn to ignore other dogs could help with this. Behavior when your dog sees another dog is an indication of how well trained they are, and how comfortable they are with the stimulation caused by encountering other dogs.
Fearful crouching dog
A dog that crouches could be fearful. Dog Training Tips says that a dog that is crouching but not exhibiting any other fear-based behavior is probably OK to approach. If you do approach a crouching dog, approach in a sort of arc rather than walking directly towards him. Do not make direct eye contact.
You may sense that the dog is afraid if he is turning his head away, yawning, or licking his lips, which are some examples of self-calming behaviors. Cases of a fearful crouching dog may be due to being under-socialized, in which case, investing some good time in bonding and training can solve the problem.