So I got this idea from the quiz site, Sporcle. In it, players had to click on an equine, feline, canine, feline. 
For example: zebra, tiger, wolf, persian until the options were used up.

I decided to bring it here, but altered for forum use.

I will name an equine. Player 2 will name a feline. 3 will name a canine and 4 will name a feline. And so on.

Acceptable: breeds, species, mythical (not one you made yourself).

SHIRE

Horse dog or cat... we've lost the pattern!!

Jackle 

I think it was correct so far? Equine, feline, canine, feline. So it would've been a feline after Magic's post. :)

So...going by that pattern,...

dalmatian

I also think it was correct? 

Siamese 

horse breed next :D

Appaloosa

Toyger 

belgian malinois

Burmese

Cleveland Bay

lion

Siberian Husky

Scottish Fold

”Sometimes people think if they love a broken person enough, they can be what finally repairs them, but the problem with that is the other person just ends up broken, too.” -Colleen Hoover

Paso Fino

Bobcat 

Alaskan Malamute

Regular moggie

Akhal-Teke

Maine coon

Dog?

Frenchie

Russian blue

Arabian