In this sentence...
彼【かれ】の挙動【きょどう】がおかしいので警官【けいかん】がつけて行【い】った
The policeman followed him because of his suspicious behavior.
https://dictionary.goo.ne.jp/word/en/挙動#je-18506
...does the verb mean "went" 行【い】った or "did" 行【おこな】った?
I found a similar example sentence in wwwjdic, but without the 行った at the end:
警官【けいかん】はその男【おとこ】をつけた。
The policeman followed him.
This would be 行く, not 行う. The latter is used to say something was carried out/executed.
Here, 行く is actually part of the format of ていく. When you add いく to a て-form of a verb, it is as if the verb is currently being acted out.
In your example, the police were likely to begin a process of following him. To say, the police began to actively follow him. It conveys a sense of progression of some event.
The other used of a directional verb you will see is てくる, which is to say some verb arrived at something, or that something was done, and arrived at that state.
太くなってきた would be something became fat after some process, to say it arrived at being fat.
