homological algebra - colimits of spectral sequences

I'm looking for some references about colimits of spectral sequences.

More precisely: let $X : I longrightarrow cal{C}$ be a functor from a filtered category $I$ to the category of double cochain complexes of an abelian category $cal{C}$, in which filtered colimits exist and commute with cohomology.

Let $E_2(X_i)$ be the second page of the first filtration ss associated to $X_i$. Assuming that the $X_i$ are right-half plane double complexes, it weakly converges to $H^*(mbox{Tot}^prod X_i)$ for all $i$ (Weibel, "An introduction to homological algebra", page 142):

$$
E_2(X_i) Longrightarrow H^*(mbox{Tot}^prod X_i) ,
$$

where $mbox{Tot}^prod$ is the total product complex,

$$
(mbox{Tot}^prod X)^n = prod_{p+q=n} X^{pq} .
$$

For the same reason:

$$
E_2(underset{i}{lim_longrightarrow} X_i) Longrightarrow H^*(mbox{Tot}^prod underset{i}{lim_longrightarrow} X_i ) .
$$

Then, because of the exactness of $displaystyle lim_longrightarrow$, we have

$$
underset{i}{lim_longrightarrow} E_2 (X_i) = E_2(underset{i}{lim_longrightarrow} X_i) .
$$

Then my question is: under which conditions can I assure that I have a comparison theorem like

$$
underset{i}{lim_longrightarrow} H^* (mbox{Tot}^prod X_i) = H^*(mbox{Tot}^prod underset{i}{lim_longrightarrow} X_i) quad mbox{?}
$$

Any hints or references will be appreciated.