  We apply differential operators to modular forms on orthogonal groups
$\mathrm{O}(2, \ell)$ to construct infinite families of modular forms on
special cycles. These operators generalize the quasi-pullback. The subspaces of
theta lifts are preserved; in particular, the higher pullbacks of the lift of a
(lattice-index) Jacobi form $\phi$ are theta lifts of partial development
coefficients of $\phi$. For certain lattices of signature (2, 2) and (2, 3),
for which there are interpretations as Hilbert-Siegel modular forms, we observe
that the higher pullbacks coincide with differential operators introduced by
Cohen and Ibukiyama.
