Yes. A Pokémon that evolves by trading in the games evolves here when you finish a real /trade with someone. If the games needed it to hold an item while traded, you use that item on it from /bag instead, and no trade is needed.
Evolve on any trade
Kadabra, Machoke, Graveler, Haunter, Boldore and Gurdurr evolve right after a completed trade. Karrablast and Shelmet only evolve when they're traded for each other, like in the games.
Use the item instead
Metal Coat, King's Rock, Dragon Scale and Up-Grade are items you use on the Pokémon from /bag. Razor Claw and Razor Fang work the same way.
Stopping an evolution
A Pokémon holding an Everstone won't evolve when traded. For level evolutions you get an Evolve or Not yet prompt, and Not yet asks again at the next level.
Stones and the rest
Evolution stones are used from /bag too: pick the stone, then the Pokémon. A Pokémon that evolves early still learns its new form's moves, and you can relearn anything it skipped from Moves in /pokemon info for ₽.