This page at the Library of Congress web site says that WHILE THE RICH MAN RIDES BY IN HIS CARRIAGE AND PAIR comes from a musical work called "The Co-optimists" (1926). Other songs from that show are:

THREE LITTLE SCHOOL GIRLS
THE PERAMBULATOR WALK
TO PASS THE TIME AWAY
ROLLING STONE
THE PIRATES LULLABY
ONCE UPON A TIME
IT'S A WONDERFUL IDEA
PIG-TAIL ALLEY
THE LONGEST DAY