Question

Once passed the Elite 4 in Pokemon X, can I make a in-game trade for the beginner that's effective against my Beginner?

???

Comments

the kanto starters?

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Well, I did say X so yeah

Answer 1

No.
You can get the starter that's weak to yours from Shauna after beating the Elite 4, but the only way to get the one that is strong against yours is to trade for it, either with someone you know, or off the GTS.