Rewriting a integral using a pullback between manifolds with different dimensions

Let $M$, $N$ be differentiable manifolds, let $f: M \to N $ be a smooth map. Let $\omega \in \Omega^{dim(N)}(N)$, a dim(N)-form on $N$. Consider the integral:

$$\int_N \omega$$

We know that in the case that $M$ and $N$ have the same dimensions and f is a diffeomorphism, we have:

$$\int_M f^*\omega = \int_N \omega$$

My question is the following: when $M$ and $N$ have different dimensions, say $dim(M) > dim(N)$ and $f$ is surjective, is there a way to express $\int_N \omega$ as a integral over $M$ using the pullback $f^*$ ?

What does make sense when $\dim M>\dim N$ is the Gysin map, usually called integration over the fiber. Assuming everything is compact, for simplicity, and oriented, given a submersion $f\colon M\to N$ and any $\phi\in\Omega^k(M)$, one has $f_*\phi\in\Omega^{k-(\dim M-\dim N)}(N)$. When $k=\dim M$, in particular, you will have $$\int_M\phi = \int_N f_*\phi.$$