Let $G$ be a Lie group, $\mathfrak g$ its Lie algebra, $K$ its maximal compact subgroup.
To every flat $G$-bundle $P$ over a smooth manifold $M$ I can associate a homomorphism $w_P: H^*(\mathfrak g, K) \to H^*(M;\Bbb R)$. This is called the characteristic homomorphism of $P$. These are natural under pullbacks in the sense that if we have a smooth map $f: M_1 \to M_2$, then $w_{f^*P_2} = f^*w_{P_2}$. ($H^*(\mathfrak g, K)$ is the Lie algebra cohomology of $\mathfrak g$ relative to the subgroup $K$ - you do the same thing as normal - coefficients in the trivial module $\Bbb R$ - but restrict to cochains that vanish on $\mathfrak k$ and are invariant under the adjoint action of $K$. If $K$ is connected this is just $H^*(\mathfrak g, \mathfrak k)$.
The homomorphism is defined as follows: let $P$ be a flat $G$-bundle over $M$; factor the projection as $P \to P/K \to M$. The connection form $\omega \in \Omega^1(P;\mathfrak g)$ defines a chain map $C^*(\mathfrak g) \to \Omega^*(P;\Bbb R)$ (defined for simplicity on 1-chains; extend the same idea in general) $\omega(\alpha)(X) = \alpha(\omega(X))$. Our restrictions on the cochains means this descends to a map $C^*(\mathfrak g,K) \to \Omega^*(P/K;\Bbb R)$. Composing now with the map induced by the homotopy equivalence $H^*(P/K;\Bbb R) \to H^*(M;\Bbb R)$ defines our characteristic homomorphism $w_P$. This process is described in pages 66-69 of Morita's "Geometry of Characteristic Classes".
Now flat bundles are classified up to isomorphism by the holonomy map $\{\pi_1(M) \to G\}/\text{conjugacy}$ in a very down-to-earth sense: if $P$ has holonomy homomorphism $\rho$, then $P \cong (\tilde M \times G)/\pi_1(M)$, where if we realize the universal cover $\tilde M$ by $$\tilde M = \{[\ell] \mid \ell: [0,1] \to M, \ell(1) = p_0\},$$ $\pi_1(M)$ acts by $\alpha \cdot ([\ell],g) = ([\ell]\alpha, \rho(\alpha)^{-1}g)$. This action preserves the trivial connection on $\tilde M \times G$, so this gives us a flat $G$-bundle over $M$.
Because our flat bundles are classified by homomorphisms $\pi_1(M) \to G$, they're classified by (homotopy classes of) maps $M \to BG^\delta = K(G,1)$, where $G^\delta$ is $G$ with the discrete topology.
In the usual theory of characteristic classes, we now demonstrate a universal bundle $EG \to BG$ that every other bundle is a pullback of, and thereby universal characteristic classes. But in our new world, $BG^\delta$ is not even close to being a manifold. We're not going to be able to write down any differential forms on it, so it's not going to make sense to talk about a flat $G$-bundle on $BG^\delta$.
Is there still a universal characteristic homomorphism $w: H^*(\mathfrak g, K) \to H^*(BG^\delta;\Bbb R)$ that pulls back to the individual characteristic homomorphisms? Do I get it for free somehow because $w$ is natural under smooth maps, and is this/is there a general phenomenon that I can appeal to: I have some sort of universality automatically coming from naturality?
(If this is not a general phenomenon, one thing I think might work is to take Sullivan's idea, work with a simplicial model of $BG^\delta$, and define smooth forms on it by doing so cell-by-cell and demanding they agree on restrictions. Maybe one can still make a theory of flat $G$-bundles in this context, perform the same classification as above, and write down a universal bundle on $BG^\delta$. I haven't thought carefully about this. Even if so, it'd be nice if there was something more general than "I can fiddle and make it work" going on here.)