  In this article we study generalised Dirac-Schr\"odinger operators in
arbitrary signatures (with or without gradings), providing a general
KK-theoretic framework for the study of index pairings and spectral flow. We
provide a general Callias-type Theorem, which allows to compute the index (or
the spectral flow, or abstractly the K-theory class) of Dirac-Schr\"odinger
operators from their restriction to a compact hypersurface. Furthermore, if the
zero eigenvalue is isolated in the spectrum of the Dirac operator, we relate
the index (or spectral flow) of Dirac-Schr\"odinger operators to the index (or
spectral flow) of corresponding Toeplitz-type operators. Our results generalise
various known results from the literature, while presenting these results in a
common unified framework.
