Here’s my attempt at getting started.

Without loss of generality, take $C = \{1, \dots k\}$.

Now for $1 \leq i \leq k$, conditional on $\mathcal F_t^C$, $X_i$ follows the dynamics

$$X^i_t= \int_0^t \lambda_i (\mathbf X_s) \, ds + w_t,$$

where now $W^i = w^i \in C[0, t]$ is deterministic given $\mathcal F^C_t$. 

As such, by taking conditional expectations, we have the equation 

$$\mathbb E[X^i_t| \mathcal F^C_t] = \int_0^t\mathbb E[\lambda_i (\mathbf X_s) | \mathcal F^C_s] \, ds+ w^i_t.$$

We can of course write similarly, for $i > k$,

$$\mathbb E[X^i_t| \mathcal F^C_t] = \int_0^t\mathbb E[\lambda_i (\mathbf X_s) | \mathcal F^C_s] \, ds + W^i_t,$$

or in SDE form,

$$d \mathbb E[X^i_t| \mathcal F^C_t] = \mathbb E[\lambda_i (\mathbf X_t) | \mathcal F^C_t] \, dt + dW^i_t.$$

I am not sure how to proceed from here. We may be able to get something much more if $\lambda$ is assumed smooth enough to apply Ito’s formula.