It certainly sounds like an interesting world, and not a prophecy I’ve heard before now.
They don’t already rule?
I, for one, welcome our new canine overlords.
They are going to be stymied by the ease with which they can be distracted by squirrels.
Leitner described a detailed narrative in which dogs slowly gained legal rights before establishing a “canine democracy” by the late 21st century. “By 2087, Labradors are judges, shepherds are presidents, and humans live as second-class citizens,” he told onlookers. “You must prepare to sit, stay, and obey.”
You have to admit that this would probably be a much healthier planet and, overall, would probably be a much better world – more just, more fun, and completely free of worry! 
Where do felines fit in this world?
Pretty sure he’s an agitator in the employ of the cats
South Park predicted a future where super-intelligent otters are at war with humans over the true nature of atheism. I agree that dogs are far more likely to rule in the near term by virture of their numbers, population diversity and distribution, lack of reliance on fresh clams and kelp, integration into human society.
I think i’m OK with it
We already have a “son of a bitch” as President.
Per Jerry Seinfeld:
If you see two life forms, one of them’s making a poop, the other one’s carrying it for him, who would you assume was in charge?
Mostly they drool.
Hear! Hear! Woof! Woof!
When my noble Bernese Mountain Dog used to go for walks, head held high in regal dignity, there would always be cats lurking behind distant bushes hissing at him. Why they hissed is a mystery since he was completely unconcerned with them and completely ignored them, but I guess felines will be felines. How do felines fit in this world of the future? Probably the same way they fit into the world today: they are irrelevant, except to themselves.
This.
You are familiar with the phrase “it’s like herding cats”? Well, then.
Alas, I will be too old to play fetch for my master by that point.
So not even Border Collies have the drive or brains to take over the world. If we’re replaced, it will probably be by raccoons who have now spread to most of Europe and have the ability to use our artifacts. They’re probably already secretly plotting and practicing in our garbage dumps.
Me, too, even now. But I was taught to play fetch some years ago. Here’s how it worked.
When my Bernese was still a growing puppy, I thought it was time to play a game of fetch. As he sat beside me I took what I think was a tennis ball and threw it across the yard. That pup then looked up at me, and I swear that his expression clearly said, “are you just throwing that ball away, or are you going to go fetch it?”
No, he was not a dumb dog – it was the opposite. He was exceptionally bright, and that was part of the problem with him. I have told stories of how he has outwitted humans. The fetch thing was minor compared to times when humans tried to outwit him, and he turned the tables on them!
Having visited Bern on more than one occasion, may I at least raise the possibility that the cats were actually saying, “SWISS! SWISSSSSS!”