Movement towards action refers to sentences like the following:
I am going to eat.
I am coming to meet her.
I went to watch a football game.
In Japanese we use the following structure.
【Verb: Stem】に【行く】or【来る】
To go/come to do something
会いに来ます。
To come and meet.
明日は、銀座に寿司を食べに行きます。
Tomorrow I’m going to go and eat sushi in Ginza [a place in Tokyo].
遊びに来ませんか?
Won’t you come and hang out.
サーカーの試合を見に行きました。
I went to watch a football game.
It is easy to slip up on the grammar here. You might think that to say “I will go to study in the library” (for example) the grammar should be…
図書館で勉強しに行きます。(✘)
I went to watch a football game.
… because the library is the place of action and the following, as we have seen, is correct:
図書館で勉強します。
I will study in the library.
However, when we want to express movement towards an action it is the final verb—“to go” or “to come”—which dictates the particle and therefore we must use に.
図書館に勉強しに行きます。
I will go to study in the library.
We need to be careful when using the directional particle.
東京駅へ行きます。(✔)
東京駅に行きます。(✔)
I will go to Tokyo station.
That’s fine. But the directional particle cannot be used to join the two verbs.