  The anomalous Hall, Nernst and thermal Hall coefficients of
Fe$_{3-x}$GeTe$_2$ display several features upon cooling, like a reversal in
the Nernst signal below $T = 50$ K pointing to a topological transition (TT)
associated to the development of magnetic spin textures. Since the anomalous
transport variables are related to the Berry curvature, a possible TT might
imply deviations from the Wiedemann-Franz (WF) law. However, the anomalous Hall
and thermal Hall coefficients of Fe$_{3-x}$GeTe$_2$ are found, within our
experimental accuracy, to satisfy the WF law for magnetic-fields $\mu_0H$
applied along its inter-layer direction. Surprisingly, large anomalous
transport coefficients are also observed for $\mu_0H$ applied along the planar
\emph{a}-axis as well as along the gradient of the chemical potential, a
configuration that should not lead to their observation due to the absence of
Lorentz force. However, as $\mu_0H$ $\|$ \emph{a}-axis is increased,
magnetization and neutron scattering indicate just the progressive canting of
the magnetic moments towards the planes followed by their saturation. These
anomalous planar quantities are found to not scale with the component of the
planar magnetization ($M_{\|}$), showing instead a sharp decrease beyond $\sim
\mu_0 H_{\|} = $ 4 T which is the field required to align the magnetic moments
along $\mu_0 H_{\|}$. We argue that locally chiral spin structures, such as
skyrmions, and possibly skyrmion tubes, lead to a field dependent
spin-chirality and hence to a novel type of topological anomalous transport.
Locally chiral spin-structures are captured by our Monte-Carlo simulations
incorporating small Dzyaloshinskii-Moriya and biquadratic exchange
interactions.
