I will welcome with open arms, but I won't chase after people who are leaving me.
来るもの拒まず去るもの追わず、は以下のように英訳できます。
I will welcome with open arms, but I won't chase after people who are leaving me.
=腕を広げて、(来る)人を迎えいれるけど、私から去る人たちは追わない
Welcome with open arms=手を広げて「人」を迎える、歓迎する
Welcome someone with open arms でもオッケーです。
I hope it helps :)