The divergence of a vector field depends on a volume form, which is a nowhere-zero $n$-form on an oriented $n$-manifold. Given a volume form, most diffeomorphisms change it, but there is a large and rich group of diffeomorphisms that preserve it. For instance,
Calabi showed (by an elementary construction) that there is a
deformation retraction from the space of all smooth diffeomorphisms on a manifold to the space of
volume-preserving diffeormorphisms --- basically because the set of choices for positively-oriented volume forms is contractible. (Note: this retraction cannot be made to preserve the group structure). I.e, you can do basically
anything you want with volume preserving vector fields except things that are obviously impossible (like a diffeomorphism of a ball to a proper subset of the ball). Quite a lot is also known about the group structure of volume-preserving diffeomorphisms,
as well, if there are questions that involve actual group relations.
There are topological and differentiable consequences of a vector field being divergence free, which are of course preserved by
diffeomorphisms. On a closed manifold, the Poincaré recurrence theorem states that for almost every x in $M^n$, the
flow line of a divergence free vector field returns infinitely often to every neighborhood of its starting point. This is because for any set of positive measure, the forward flow-tube builds up volume at a steady rate, so if there's an upper bound, it has to come back and intersect itself.
The diffeomorphic image of a divergence free vector field has the property that it can be multiplied by a smooth function --- that is, the speed of the flow can be regulated --- to make it divergence free again. Given a diffeomorphism, just multiply by the ratio of the volume in the domain to the volume in the range. I.e., just like in a stream, the flow slows down when it spreads out and has to speed up to maintain flow through a small gap.
Divergence free vector fields are in 1-1 correspondence with closed $n-1$ forms, by the correspondence
$$X \leftrightarrow i_X(\Omega), $$
where $X$ is the vector field, $\Omega$ is the volume form, and $i_X$ means stick $X$ in the first slot of $\Omega$, thought of as a function from an n-tuple of vectors to the real numbers. You can think of $i_X(\Omega)$ as the flux or flow rate of the vector field $X$ through an infinitesimal element of $n-1$-dimensional area. If $X$ is the instantaneous flow of the Mississippi river in time units of second, and you integrate $i_X(dV)$ on a surface that cuts across the entire river, you get the amount of water per second flowing through the Mississippi at that place.
It's easy to work with closed $n-1$ forms, by
starting with a basis for cohomology, then modifying by $d(n-2$ form$)$. You can take a diffeomorphism that is the time t flow of a divergence free vector field, and it will preserve divergence-freeness of any other vector field. This does not get all
examples even near the identity, but by integrating time-dependent families of divergence free vector fields, you get arbitrary volume-preserving vector field that are isotopic to the identity, and containing a neighborhood of the identity.