Please explain the difference between these words

それで、中のほうへ（＿＿＿）が、行けませんでした。
1）行くようになりました
２）行こうとしました
３）行かせました
４）行かれました

ドアのところに立っている人たちが（＿＿＿）からです。
１）動かせた
２）動かれた
３）動くようになった
４）動こうとしない

Could someone please explain the meaning of these answers.

When asking these questions, it'd be helpful if you could post your best attempt at answering, along with specific questions about what you don't understand.
If we just give you the answers, it's unlikely to be very helpful.

The first sentence is establishing a contrast. The speaker tried to get to the center, but couldn't get there. Only one of the choices given fits with that meaning of contrast. Any of the other three would result in a nonsensical statement.

Question #2, I'm assuming, is meant to follow Question #1. Thus, it's providing the reason that the speaker wasn't able to get in. Again, only one of the choices given captures that meaning. The others would all seem to imply the opposite of this.

Care to take a try at answering it now?

In addition, you need to give us the context where those sentences are used to get the correct answers. Those are a part of a paragraph that describes a situation where the writer was trying to get on a train, perhaps? If so, the correct answer is only one for each question, as jt_-san explained.

However, if those are stand alone sentences, other answers can be grammatically correct in appropriate contexts. I know questioners don't make those kinds of questions, but you need to realize that multiple answers can be correct depending on the situation and it's important to show the context when asking questions.