The surname Swallow is English in origin. According to ancestry.com and the Dictionary of American Family Names the Swallow surname is thought to come from "a nickname for someone who resembles a bird". It could be locational in meaning from the Swallow river near Lincolnshire, England.
I only have one Swallow, so far, in my family tree.
Mary Ann Swallow
Birth: 7 August 1817 New York
Death: 10 September 1875 Adrian, Lenawee, Michigan
Mary Ann Swallow married William G. Dyer sometime before 1838, when their first child, Adda Dyer was born.
I haven't spent any time on this line, so I don't consider it a brickwall.