  In this paper we study the electroproduction of open heavy flavor $D$- and
$B$-mesons in the kinematics of future $ep$ colliders, such as the Electron Ion
Collider (EIC), the Large Hadron electron Collider (LHeC) and the Future
Circular Collider (FCC-he). We study in detail the dependence of the
cross-sections on multiplicity of co-produced hadrons, in view of its possible
sensitivity to contributions from multipomeron contributions, and discuss
different observables which might be used for its study. According to our
theoretical expectations, in $ep$ collisions the multipomeron contributions are
small in the EIC kinematics, although they might be sizable at LHeC and FCC-he.
We also provide theoretical predictions for the production cross-sections of
heavy mesons in the kinematics of all the above-mentioned $ep$ colliders.
