Suppose we are working over CW complexes. Are there any nontrivial natural transformations $H^n (-;G) \rightarrow H^k(-;G)$ if $n \not= k$. I believe I can answer the question for $n=1$ and $G=\mathbb{Z}$.
By the representation theorem for cohomology of CW complexes, $H^1 (-;G)$ is represented by the circle and $H^k (-;G)$ is represented by $K(G,k)$. We know natural transformations between these functors (considered as set valued) correspond to basepointed homotopy classes of maps from the circle to $K(G,k)$. This means the natural transformations (as set valued functors) are in correspondence with $\Pi_1(K(G,k))$, so if $k>1$ there is only the trivial transformation sending everything to the identity. If $k=1$, these natural transformations should correspond to the homomorphism $x \rightarrow nx$ for all integer $n$.
For higher $n$ and any $G$ it seems like we will have a natural correspondence between the sets $\langle K(G,n),K(G,k)\rangle=H^k(K(G,n))$ and natural transformations between our functors as set valued functors. A little thinking should show that in general if $k>n$ there are no natural transformations, but the obvious argument relies on the Universal Coefficient Theorem. I've heard that the cohomology of Eilenberg-MacLane spaces is known, so I have a couple questions:
Does each element of the cohomology necessarily correspond to a natural transformation whose components are homomorphisms?
Are there nice descriptions of some of these natural transformations if $n>1$ and $G=\mathbb{Z}$?
Is it possible to go the other way around and get at the cohomology of Eilenberg-MacLane spaces through these natural transformations? I think for the case $n=1$ it should be possible to prove that these are the only natural transformations directly.