on one's way
See on the way.
Also, be on one's way. Leave, get going, as in “On your way,” said the officer, trying to move the crowd, or It's been a wonderful party but we must be on our way now. [Early 1900s]
The American Heritage® Idioms Dictionary Copyright © 2002, 2001, 1995 by Houghton Mifflin Harcourt Publishing Company. Published by Houghton Mifflin Harcourt Publishing Company.