YOU SAID:

Mother went missing to find a cat

INTO JAPANESE

母が猫を探しに行方不明になりました

BACK INTO ENGLISH

My mother went missing looking for her cat.

INTO JAPANESE

母が猫を探して行方不明になりました。

BACK INTO ENGLISH

My mother went missing while looking for her cat.

INTO JAPANESE

母が猫を探している途中で行方不明になりました。

BACK INTO ENGLISH

My mother went missing while looking for her cat.