The distance and time your puppy should walk for, varies according to the breed. So it is advisable you always watch your dog whenever you take him on a walk and try to notice when he is tired probably when his speed reduces, etc. When you notice he is tired, Stop the walk in order not to Over-exercise him.
Depending slightly on the breed, a puppy’s skeleton doesn’t fully mature, and the growth plates do not fuse until the puppy is 18 months old or older. The growth plates, which are part of the bone structure, typically fuse during the puppy’s tenth month, but after that the skeleton continues changing as the bones adapt to the various kinds of stress put on them. The skeleton is not fully formed and bones are not hardened until 18 months or later.
Before the growth plates close, too much exercise can jar and loosen little joints, and cause stress to the skeleton, which can cause permanent damage including elbow and hip dysplasia, joint problems, and growth abnormalities in the cartilage that can cause pain and lameness. (Source : How Far and How Long Should You Walk Your Puppy?)
However, as pointed out by My dog’s life,
There is a 5 minute rule which is basically 5 minutes walk per day, per month of your puppy’s life, so therefore a 3 month old puppy can be walked for 15 minutes and a 4 month old puppy for 20 minutes, which is a guideline but as an owner make your own judgement on your puppies needs as each puppy is different.
It is important to walk your puppy each day to help with training, plus during a walk he/she will gain social skills, mental stimulation and physical exercise. As your puppy grows you can extend the time of each walk and remember to go on different walks with new surrounding and scents which will be more stimulating for your puppy, plus more enjoyable for you as you will meet new dog walkers during your walk.
In addition, when you want to take him for a walk, it is advisable you go to a clean environment free of dirt, dog poop, etc. Also you can take along with you a pedometer so that you can measure the amount of steps your puppy took before he grew tired. Then you can apply this to your other walks.
Because puppies’ bones are still growing, most experts recommend that puppies are not taken on long walks until they are approaching their first birthday.
An often quoted rule of thumb is five minutes of exercise per day for every month of the puppy’s age. This would mean no more than 35 minutes of walking a day for a seven month old dog.(source)
Another thing to consider is a the breed of the dog. Smaller dogs usually can not go as far as a dog like a lab can. And the larger breeds take more stress on their joints making long walks somewhat dangerous to them.
The amount of exercise your puppy needs depends on his age, breed and medical condition. Not every breed will be up for a long walk through Central Park. Some breeds are just not built to go the distance, while others are always ready to romp. Some of the breeds that need the most exercise are Border collies, Labrador retrievers, Jack Russell terriers and Dalmatians. (source)
Be sure that your dog does not overheat.
If you are too hot, assume your puppy is too hot, too. Provide water to your puppy or try to run in a cooler area.
Be consistent with exercising puppy.
The weekend marathon after a week of couch potato life is very hard on the joints.
Be cautious with your puppy’s feet.
Running on cement (especially hot cement) can cause sloughing of the foot pads, so try to have breaks on softer surfaces or at least work your puppy up to the harder surfaces to give them a chance to form the necessary calluses.
Be aware of your puppy’s limits.
Until you understand your puppy’s stamina, be sure that you are able to stop when your puppy starts to lag behind. That means taking the short route 5 times instead of the long route once. If you feel that the puppy is not able to go as far as he should, consult a vet to see if there is a health reason for the intolerance.