  We investigate how the consistency relations of large-scale structures are
modified when the initial density field is not Gaussian. We consider both
scenarios where the primordial density field can be written as a nonlinear
functional of a Gaussian field and more general scenarios where the probability
distribution of the primordial density field can be expanded around the
Gaussian distribution, up to all orders over $\delta_{L0}$. Working at linear
order over the non-Gaussianity parameters $f_{\rm NL}^{(n)}$ or $S_n$, we find
that the consistency relations for the matter density fields are modified as
they include additional contributions that involve all-order mixed
linear-nonlinear correlations $\langle \prod \delta_L \prod \delta \rangle$. We
derive the conditions needed to recover the simple Gaussian form of the
consistency relations. This corresponds to scenarios that become Gaussian in
the squeezed limit. Our results also apply to biased tracers, and velocity or
momentum cross-correlations.
