I don’t specialize in animal portrait, but I’ve done a few that I can share some tips on how I do it.

The images below are taken outdoor, so I’ll focus on how I did this session.
Dogs are hard to control as they usually run wildly once they are unleashed. That’s ok, we can start off with some photos capturing them running.
There are two situations; 1) The dog runs parallel with your camera. That is when they run across from left to right or right to left. 2) The dog runs toward or outward from your camera. The difference between the 2 situations is that for 1), the distance between doggie and you is constant whereas 2), the distance changes dramatically.
So the hard part is to capture the doggie in focus! For 1), since the distance is constant, the auto focus on the camera is pretty reliable. Keeping the shutter speed above 1/500 and/or aperture of f2.8 will get you many sharp images. You could also pan the dog and keep the dog within the frame while pressing the shutter. But the panning effect won’t be very noticeable as the shutter speed is too fast. So if you want panning effect. Slower the shutter speed to 1/125. I won’t go into too much detail of panning since I didn’t do that on my session.
For 2), when the dog is sprinting towards you, how can you get sharp images? Basically, unless you have top of the range camera body and lenses. Your auto focus is not capable of getting sharp images. Trust me, I used Nikon D700 and prime lens, but AF didn’t go too well. So what I did was I set manual focus and frame the location that I want to capture the dog and when the doggie runs into that spot, I press the shutter with the continuous mode turned on. So I ended up with at least 2-3 sharp images at aperture f4. With the dog running towards you a few times, you should have at least 10 nice sharp images.
I worked with a prime lens of 85mm. With this focal length, I could include more landscapes into the image. With a lens of 200mm and longer, you could get images of the dog fill up the frame.

After some running session, you and the dog probably needs to get some rest. So now is time to take some still photos. The doggie would spend more time sitting or lying. So lens selection can be anything you like. For me, I would use my 50mm to get some detailed photos on eyes, nose, feet and name badge. Even when the doggie moves a little, you can still manage the distance.
While the doggie is sitting or standing, to get him to look at the camera is also tricky. Dogs are really sensitive with sound, so they will get distracted and look away easily. What I did with my session was to make some strange noise that would keep his attention. This is all up to your creativity, there’s no standard sound that would keep your dog’s attention.

So that’s how I did my session. Not to mention that lighting is always important. I take photos 1-2 hours before sunset, so I don’t get harsh lights and the dog don’t get exhausted too easily. After all, it’s to enjoy the outdoor session.