Julie Bergner defined the homotopy pullback of a diagram of model categories, in Homotopy fiber products of homotopy theories, and Homotopy limits of model categories and more general homotopy theories. In the case of homotopy pullbacks, the construction appeared a little bit earlier in Bertrand Toen's Derived Hall Algebras. Given a diagram of model categories and left Quillen functors
\begin{align} M_1 \stackrel{F_1}{\to} M_3 \stackrel{F_2}{\gets} M_2 \end{align}
Bergner defines the homotopy pullback to be a category $M$ whose objects are 5-tuples $(x_1,x_2,x_3,u,v)$ with $x_i \in M_i$ and $F(x_1) \stackrel{u}{\to} x_3 \stackrel{v}{\gets} F(x_2)$. Morphisms are the obvious thing. This category $M$ can be given a model structure where the weak equivalences and cofibrations are levelwise (on each $x_i$), and that model structure can be localized if desired to force $u$ and $v$ to be weak equivalences in the local objects of $M$. Bergner then proves $M$ has the correct homotopy type, meaning that, upon passage to complete Segal spaces, it becomes the actual homotopy pullback of the diagram. It appears that there is nothing special about $F_i$ being left Quillen, and you could make an analogous construction of the homotopy pullback of a diagram of right Quillen functors.
Here's my question:
(1) In a homotopy pullback diagram of right Quillen functors, if $F_2$ is a right Quillen equivalence, is the same true for the induced functor from the homotopy pullback $M$ to $M_1$?
Dually, consider a homotopy pushout of left Quillen functors
$$ \begin{array}{ccc} M_3 & \xrightarrow{F_1} & M_1 \newline \downarrow & & \downarrow \newline M_2 & \xrightarrow[F_1']{} & P \end{array} $$
$P$ is defined in Homotopy Colimits of Model Categories, where objects are 5-tuples with $F_1(x_3) \to x_1$ in $M_1$ and $F_2(x_3)\to x_2$ in $M_2$, with weak equivalences objectwise. Then:
(2) If $F_1$ is a left Quillen equivalence, will $F_1'$ give an equivalence of homotopy theories?
