  The fundamental optical excitations of correlated transition-metal compounds
are typically identified with multielectronic transitions localized at the
transition-metal site, such as $dd$ transitions. In this vein, intense interest
has surrounded the appearance of sharp, below band-gap optical transitions,
i.e. excitons, within the magnetic phase of correlated Ni$^{2+}$ van der Waals
magnets. The interplay of magnetic and charge-transfer insulating ground states
in Ni$^{2+}$ systems raises intriguing questions on the roles of long-range
magnetic order and of metal-ligand charge transfer in the exciton nature, which
inspired microscopic descriptions beyond typical $dd$ excitations. Here we
study the impact of charge-transfer and magnetic order on the excitation
spectrum of the nickel dihalides (NiX$_2$, X $=$ Cl, Br, and I) using Ni-$L_3$
resonant inelastic x-ray scattering (RIXS). In all compounds, we detect sharp
excitations, analogous to the recently reported excitons, and assign them to
spin-singlet multiplets of octahedrally-coordinated Ni$^{2+}$ stabilized by
intra-atomic Hund's exchange. Additionally, we demonstrate that these excitons
are dispersive using momentum resolved RIXS. Our data evidence a
ligand-mediated multiplet dispersion, which is tuned by the charge-transfer gap
and independent of the presence of long-range magnetic order. This reveals the
mechanisms governing non-local interactions of on-site $dd$ excitations with
the surrounding crystal/magnetic structure, in analogy to ground state
superexchange. These measurements thus establish the roles of magnetic order,
self-doped ligand holes, and intersite coupling mechanisms for the properties
of $dd$ excitations in charge-transfer insulators.
