By standard deformation theory (see e.g., Hartshorne III Ex 4.10, but there are probably better references), the tangent sheaf of $M_g$ is $R^1\pi_{\ast}(C, T_{C/M_g})$, which is Serre dual to $\pi_{\ast}F$. The tangent sheaf is dual to what you wanted.

By standard deformation theory (see e.g., Hartshorne III Ex 4.10, but there are probably better references), the tangent sheaf of $\mathscr{M}_g$ is $R^1\pi_{\ast}(\mathscr{C}, T_{\mathscr{C}/\mathscr{M}_g})$, which is Serre dual to $\pi_{\ast}\mathscr{F}$. The tangent sheaf is dual to what you wanted.

By standard deformation theory (see e.g., Hartshorne III Ex 4.10, but there are probably better references), the tangent sheaf of M_g is R¹pi_*(C, T_C/Mg), which is Serre dual to pi_*F. The tangent sheaf is dual to what you wanted.

By standard deformation theory (see e.g., Hartshorne III Ex 4.10, but there are probably better references), the tangent sheaf of $M_g$ is $R^1\pi_{\ast}(C, T_{C/M_g})$, which is Serre dual to $\pi_{\ast}F$. The tangent sheaf is dual to what you wanted.