Is anything known about the maps out of an Eilenberg Mac-lane Space $K(G,n)$?

Obviously I'm interested in extensions of Miller's resolution of the Sullivan conjecture, that $Map_*(K(G,1),X)\simeq\ast$ for $G$ a discrete, locally finite group and $X$ a connected, finite complex. On the other hand Gray has shown that there are uncountably many phantom maps $K(\mathbb{Z},2)\rightarrow S^3$ so it seems like there is nothing much to say without some specialisation. So allowing for some method of getting rid of phantom maps, say, p-completion, rationalisation, etc... is anything known?

Recently there have started to emerge some novel applications for the nullificiation and cellularisation functors of Dror Farjoun in the context of $BZ_p$-null homotopy theory. I'd like to know what $K(\mathbb{Z},2)$-, $K(\mathbb{Z}_p,2)$- and $K(\mathbb{Z},3)$-null homotopy theory looks like.