More than that: they were "yellow bounders"!

Must be quite an experience, a "yellow bounder".
Just looked it up at the Glossary page of Regency World.

So economical in design is the "yellow bounder,"
that the person in charge rides on the back of the horse,
or one of the horses, pulling the vehicle;
there is no coachman's or driver's seat on top of the thing.

The vehicle is classed as a "post-chaise"
and it is rarely owned, but rather
hired from posting inns, complete with
"post boys" who were generally too old to actually be boys.

Goodness, the things I learn while looking stuff up.