Well Eevee General said it'd be good to speculate on the abilities. As Game Freak is usually flavorwise but goes to the competitive side at times, these are some predictions:
Mr Mime: I think this can go two ways. Either it gets an ability like Magic Bounce, as basically summons Magic Coat all the time, which does the same as Magic Bounce. That set, while it fits with his movement, it doesn't really fit with its other design. Also Umbreon has an arrow to it so that would mean Synchronize. Again an ability that'd kinda fit due to its psychic powers, but not specifically. Kinda useless though.
Another way it could go down is like Telepathy, a pretty useless ability that seems to fit designwise. I don't know why, it does with a lot of Psychic types. It could get some new ability that would show its ability to make invisible walls really well.
Ivysaur, Squirtle and Charizard
I guess these could get the abilities of the Pokemon listed at them. Ivysaur could get Effect Spore and Dry Skin. They are decent flavorwise as it has a huge bulb and is basically a huge leaf, but not amazing as Ivysaur doesn't specifically spread spores or has a stench (I know it gets the spore moves, but it's not known for it). Dry Skin would be strange as it's known for being boosted in the sun... This is excluding their own hidden abilities as they'll probably be changed too. That said they might shift so idk
Squirtle could get Rough Skin and... Speed Boost. Speed Boost would obviously be great, but as I said last time, I don't believe they want to transfer HA, so it's probably not that. I guess Rough Skin could fit, but it's not really known for it and all it has is a rough shell. I don't know about this one tbh.
Charizard could get Intimidate and Shed Skin. Intimidate would work flavorwise as its huge and threatening and great competitively as it can use it pre-Mega evolution, which is great. I hope it becomes this. Shed Skin could work as it's a lizard but it doesn't really look like it can shed its skin, so I don't think that'll be the ability.
Espeon could get Wonder Skin as Venonat and Venomoth are close to it. It'd have to follow the shifting mechanic but to be honest.. Compoundeyes and Tinted Lens don't fit, though Tinted Lens would be incredible. Run Away from Venonat would be possible too, but mediocre. It does fit as it's a quick cat, sorta.
Eevee is shown in Poke Amie with arrows. Does this mean Espeon and Umbreon become Poke Amie evo's too based upon the time of the day?
That's it for my speculations as most others don't seem to fit, though notice that all are gen 1 or 2.