entering training arena of wrong specialty

Is there any serious disadvantage in training a horse at the arena which spacialty doesn't match horse's specialty ?

^{There is no disadvantage to training in wrong speciality arenas. If you train at the right speciality and grade, you have a boost of higher exp gained.  If you train at wrong speciality arenas you get less. i.e I got +200exp for training a horse in the right grade and discipline, and +36exp for the same level but wrong discipline arena.} 

Someone correct me if I'm wrong but I think the horse itself - as training progress and stat boost point progress - isn't affected right? I know they still get their stat boosts but I've never checked whether training progress is less or not.

Thank you, maple !

^{They will always gain one stat boost per training session. I'm not sure if the experience for level is affected but all my horses hit level 5 training when they hit 12 years old if I don't miss any session so I think it's linear no matter what arena you train in.} 

It doesn't matter what arena you train them in for the training boost points, the only advantage to training in the right discipline and right grade is that you receive a greater amount of player exp points for leveling up your account.

So
If you put a horse in the correct grade and discipline your horse gets a training boost point, and you get the max amount of player exp from training

If you train a horse in the right grade but wrong discipline you'll get the horse training boost point and less player exp

If you train a horse in the wrong grade but right discipline you'll get a horse training boost point and the same amount of player exp as if you entered the right grade and wrong discipline (less than if you trained in right grade and discipline)

If you train a horse in the wrong grade AND wrong discipline then you still get one horse training point, but you don't get any player exp

As far as I know, what training arena you put a horse in does not effect the quantity of horse stats gained when using boost points from training

Thank you for this in depth explanation - that's super helpful

Boost points is also linear depending on your horse's grade level when using them! +11 points for Novice, +13 Local, +15 regional, +19 National, and +23 International. The amount in each discipline area is random, but the total always is the same.