Let $M$ be an oriented, compact smooth $d$-dimensional manifold with boundary, and let $\omega_t$ be a smooth family of $d$-forms that agree on some open neighbourhood of $\partial M$.
(i.e $\omega_t=\omega_0$ for all $t$ in a neighbourhood of $\partial M$. )
Since $\omega_t$ are closed, they are exact, so there is a corresponding family of $d-1$ forms $\eta_t$ such that $d\eta_t=\omega_t$.
Can we always choose $\eta_t$ in such a way that they all agree on $\partial M$?
No.
Take $M=[0,1],$ and let $f:M\to\mathbb{R}$ be a bump function. Namely, $f$ is non-negative, positive in a neighborhood of $1/2$ and vanishes on a neighborhood of the boundary. Now take $\omega_t=tfdx$. If $\eta_t$ is a family of functions with $d\eta_t=\omega_t$, then $$\eta_t(1)-\eta_t(0)=\int_0^1\omega_t=t\int_0^1fdx,$$ and the expression on the right depends on $t$.
Remark: This example can be easily generalized to arbitrary dimension. Indeed, if you have a family $\eta_t$ of $d-1$-forms which agree on the boundary, Stokes' theorem tells you that $\int_M\omega_t$ is independent of $t$.
