We characterize sequence-covering (resp., 1-sequence-covering,inebreak 2-sequence-covering) mssc-images of locally separable metric spaces by means of $\sigma$-locally finite $cs$-networks (resp., $sn$-networks, $so$-networks) consisting of $\aleph_0$-spaces (resp., $sn$-second countable spaces, $so$-second countable spaces). As the applications, we get characterizations of certain sequence-covering, quotient mssc-images of locally separable metric spaces.