  Periodicity of long wavelength moir\'e patterns is very often destroyed by
the inhomogeneous strain introduced in fabrications of van der Waals layered
structures. We present a framework to describe massive Dirac fermions in such
distorted moir\'e pattern of transition metal dichalcogenides homobilayers,
accounting for the dynamics of layer pseudospin. In decoupled bilayers, we show
two causes of in-plane layer pseudospin precession: By the coupling of layer
antisymmetric strain to valley magnetic moment; and by the Aharonov-Bohm effect
in the SU(2) gauge potential for the case of R-type bilayer under antisymmetric
strain and H-type under symmetric strain. With interlayer coupling in the
moir\'e, its interplay with strain manifests as a non-Abelian gauge field. We
show a genuine non-Abelian Aharonov-Bohm effect in such field, where the
evolution operators for different loops are non-commutative. This provides an
exciting platform to explore non-Abelian gauge field effects on electron, with
remarkable tunability of the field by strain and interlayer bias.
