dg.differential geometry - Twisting an L-infinity-morphism with "non-associated" Maurer-Cartan elements

Background

Suppose we are given $L_infty$-algebras $(g,Q)$ and $(g',Q')$ and an $L_infty$-morphism $F$ from $(g,Q)$ to $(g',Q')$. Furthermore, we have a Maurer-Cartan element $pi$ of $(g,Q)$.
One can twist $(g,Q)$ with the Maurer-Cartan element $pi$ and obtains a new $L_infty$-algebra that we call $(g,Q_pi)$. Furthermore, we can construct a Maurer-Cartan element $pi'$ of $(g',Q')$ by the formula

$pi' = sum_{n=1}^infty frac{1}{n!} F_n(pi, ldots , pi)$,

where $F_n$ is the $bigwedge^n g rightarrow g'$-part of $F$. I don't know whether there is a (better) term, so I call $pi$ and $pi'$ associated Maurer-Cartan elements

One can twist the morphism $F$ with the Maurer-Cartan elements $pi$ and $pi'$ and obtain an $L_infty$-morphism $F_pi$ from $(g,Q_pi)$ to $(g',Q'_{pi'})$. The references I found are
Dolgushev: A Proof of Tsygan's Formality Conjecture for an Arbitrary Smooth Manifold (section 2.4) and Yekutieli: Continuous and Twisted L_infinity Morphisms
(section 3).

Question

Given $L_infty$-algebras $(g,Q)$ and $(g',Q')$, an $L_infty$-morphism $F$ from $(g,Q)$ to $(g',Q')$, a Maurer-Cartan elements $pi$ of $(g,Q)$ and a Maurer-Cartan element $omega$, $omeganeq pi'$, of $(g',Q')$. Can one construct an $L_infty$-morphisms between $(g,Q)$ twisted with the Maurer-Cartan element $pi$ and $(g',Q')$ twisted with the Maurer-Cartan element $omega$, where the Maurer-Cartan elements are not "associated"? I.e. can one construct an $L_infty$-morphism between $(g,Q_pi)$ to $(g',Q'_omega)$?