As performers and teachers, we often deal in visual imagery to help evoke the right physical response.
When I'm playing my slow lip slurs, I want to maintain an unbroken airstream as I move from note to note. It helps me to think of my embouchure as
SUPPLE AND MALLEABLE instead of
BRITTLE.
For the air during these lip slurs, I like to think of an unbroken stream of air connecting the notes. Like that taffy-pulling machine above..... (or below)...