Here’s my attempt at getting started. The following answer concerns the easier case of conditioning on $\sigma(\{W^i\}_{i \in C})$.
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
$$X^i_t= X^i_0 + \int_0^t \lambda_i (\mathbf X_s) \, ds + w_t,$$
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
$$\mathbb E[X^i_t| \mathcal F^C_t] = X^i_0 + \int_0^t\mathbb E[\lambda_i (\mathbf X_s) | \mathcal F^C_s] \, ds+ w^i_t.$$
We can of course write similarly, for $i > k$,
$$\mathbb E[X^i_t| \mathcal F^C_t] = X^i_0 + \int_0^t\mathbb E[\lambda_i (\mathbf X_s) | \mathcal F^C_s] \, ds + W^i_t,$$
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.