One family is from Siemianowka, Poland, near Bialystok (Rumsziszker, later Rubinstein). Another is from Vilna (Rumszycki, later Rumschitzki).
From Olga Zabludoff -
I found a short biography on Joseph M. Rumshisky at the LOC. He was born in Vilna 1879-81 (exact year unknown). He grew up in a musical environment. His father sometimes led the synagogue services, his mother organized vocal groups. Joseph arrived in the US in 1904 via London. He composed over 100 operettas and operas for the Yiddish theater. He also composed popular songs and liturgical music.
In 1944, 12 years before his death, he wrote an autobiography in Yiddish entitled "Klangen fun Mein Leben" (Echoes of My Life).
My conclusion is that Joseph's father, grandfather or great-grandfather was born in Rumshishok, for how else would one get the surname Rumshisky? When this ancestor of Joseph moved to Vilna, he became known as Rumshisky, (or "from Rumshishok"). The family may not have had a surname yet when they lived in Rumshishok.
Come to think of it, there was even a family named Rumshishker living in Butrimonys in 1858.