  We show that for dual-unitary kicked chains, built upon a pair of complex
Hadamard matrices, correlators of strictly local, traceless operators vanish
identically for sufficiently long chains. On the other hand, operators
supported at pairs of adjacent chain sites, generically, exhibit nontrivial
correlations along the light cone edges. In agreement with Bertini et. al.
[Phys. Rev. Lett. 123, 210601 (2019)], they can be expressed through the
expectation values of a transfer matrix $T$. Furthermore, we identify a
remarkable family of dual-unitary models where an explicit information on the
spectrum of $T$ is available. For this class of models we provide a closed
analytical formula for the corresponding two-point correlators. This result, in
turn, allows an evaluation of local correlators in the vicinity of the
dual-unitary regime which is exemplified on the kicked Ising spin chain.
