The compositional technique you're referring to is likely "imitation." In music, imitation occurs when one instrument or voice plays or sings a motif or phrase that is similar to that of another instrument or voice, often shortly after the first. This technique is commonly used in various musical styles, especially in genres such as counterpoint and polyphony, where ideas are exchanged between different parts.
Another related term is "canon," where one voice or instrument exactly mimics another after a certain time interval. Both techniques are effective in creating interplay and texture within a piece of music.