I want to make videos with different fps match the playback speed.

動画はフレームレート以外にインターレースの有無に違いはありますか。
また、動画のコーデックは何でしょうか？

プレビューが正常ならば、レンダリングするコンポジションを新たなコンポジションにネスト化してからレンダリングしてみてはどうでしょう。

The frame rate of the composition has nothing at all to do with the playback speed of the clip. Playback speed is determined by the interpreted frame rate of the original clip. You can check that by selecting the video in the Project Panel and using the File>Interpret Footage>Main Menu. If the interpreted frame rate is incorrect and you know the correct frame rate you can change it. Almost all of the time the interpreted frame rate is correct.

Drop a 60 fps clip that is 10 seconds long in a comp that is 10 fps and the footage will still be 10 seconds long, you just won't see all of the frames. Drop a 30 fps clip that is 10 seconds long in a 60 fps comp and you will see every frame twice and the clip will still be 10 seconds long. 

I hope this clears that up. If you are having speed problems then the interpreted frame rate of the footage is wrong or the footage was shot with a variable frame rate that After Effects cannot handle.