  Given a compact Riemannian manifold with boundary, we prove that the space of
embedded, which may be improper, free boundary minimal hypersurfaces with
uniform area and Morse index upper bound is compact in the sense of smoothly
graphical convergence away from finitely many points. We show that the limit of
a sequence of such hypersurfaces always inherits a non-trivial Jacobi field
when it has multiplicity one. In a forthcoming paper, we will construct Jacobi
fields when the convergence has higher multiplicity.
