A complex Mixed Hodge Structure is given by a complex vector space $V$ together with a descending filtration $W$ and two ascending filtrations $F,\bar{F}$ that satisfy the condition

\begin{equation} gr_F^pgr_{\bar{F}}^qgr_n^W(V)=0\qquad\text{if }n\neq p+q \end{equation}

It is a result of Deligne that this is the equivalent to the data of a bigraded vector space $\tilde{V}$, together with an automorphism $\delta$ satisfying

\begin{equation} (\delta-1)(\tilde{V}^{p,q})\subset\bigoplus_{p'<p,q'<q}\tilde{V}^{p',q'} \end{equation}

Is there a version of this that carries over to families, i.e. can a variation of Mixed Hodge Structure also be described by something like a (maybe $C^{\infty}$-) bundle with a decomposition into subbundles and a bundle automorphism? If yes, how does Griffiths transversality enter the picture?

Remark: I posted this on MSE before, where it was not answered.