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In this work, we consider the full Horndeski Lagrangian applied to wormholegeometries and present the full gravitational field equations. We analyse thegeneral constraints imposed by the flaring-out conditions at the wormholethroat and consider a plethora of specific subclasses of the HorndeskiLagrangian, namely, quintessence/phantom fields, $k$-essence, scalar-tensortheories, covariant galileons, nonminimal kinetic coupling, kinetic gravitybraiding, and the scalar-tensor representation of Gauss-Bonnet couplings,amongst others. The generic constraints analysed in this work serve as aconsistency check of the main solutions obtained in the literature and drawsout new avenues of research in considering applications of specific subclassesof the Horndeski theory to wormhole physics.

General constraints on Horndeski wormhole throats