  We present an advanced first-principles formalism to evaluate the
Dzyaloshinskii-Moriya interaction (DMI) in its modern theory as well as Berry
curvatures in complex spaces based on a higher-dimensional Wannier
interpolation. Our method is applied to the Co-based trilayer systems
Ir$_\delta$Pt$_{1-\delta}$/Co/Pt and Au$_\gamma$Pt$_{1-\gamma}$/Co/Pt, where we
gain insights into the correlations between the electronic structure and the
DMI, and we uncover prominent sign changes of the chiral interaction with the
overlayer composition. Beyond the discussed phenomena, the scope of
applications of our Wannier-based scheme is particularly broad as it is ideally
suited to study efficiently the Hamiltonian evolution under the slow variation
of very general parameters.
