= Solution
<Moser's trick> turns variation of <symplectic forms> into an equation for a time-dependent <vector field>. Let $\omega_t$, $0\leq t\leq1$, be a smooth path of <symplectic forms> on a compact manifold without boundary, with a constant <de Rham cohomology> class. Choose a smooth family of one-forms $\alpha_t$ such that $\dot\omega_t=d\alpha_t$. Such a smooth choice can be made using a fixed auxiliary metric; the essential requirement is this exactness throughout the path.
Nondegeneracy uniquely determines $X_t$ by
$$
\iota_{X_t}\omega_t=-\alpha_t.
$$
Let $f_t$ be its flow, with $f_0=\mathrm{id}$. Compactness ensures existence over the whole parameter interval. By <Cartan's magic formula>,
$$
\frac d{dt}f_t^*\omega_t
=f_t^*(\dot\omega_t+\mathcal L_{X_t}\omega_t)
=f_t^*(d\alpha_t+d\iota_{X_t}\omega_t)=0.
$$
Thus
$$
\boxed{f_t^*\omega_t=\omega_0.}
$$
The path is made constant by a <diffeomorphism> moving with $X_t$. Every interpolating form must be nondegenerate; equal endpoint cohomology alone does not ensure that every linearly interpolated form is a <symplectic form>. On a noncompact manifold one instead needs completeness of this flow, or restricts to a sufficiently small neighborhood, as in the local argument below.
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