Suppose $E\simeq\text{hocolim}_iE_i$ is a filtered homotopy colimit. Suppose $X=\{X_j\}_j$ is a prospace (assume some finiteness conditions on the spaces or spectra if you have to...for example, assume $X_i$ are simplicial sets with finitely many simplicies in each degree). Is it true that in the $\infty$category of prospectra, we have $\text{hocolim}^{pro}_i\{E_i\wedge X_j\}_j\simeq \{\text{hocolim}_i(E_i\wedge X_j)\}_j$? More generally, is it true that if $E_i$ are prospectra with some mild finiteness condition and all indexed by the same cofiltered diagram, then I can compute their homotopy colimit degreewise?
This is not true, unfortunately. Here's an example.
Let $E_i$ be the directed system $$ \DeclareMathOperator{\hocolim}{hocolim} \DeclareMathOperator{\holim}{holim} S^0 \to S^0 \vee S^1 \to S^0 \vee S^1 \vee S^2 \to \dots $$ of wedge inclusions, and let $X_j$ be the inverse system $$ S^0 \leftarrow S^0 \vee S^1 \leftarrow S^0 \vee S^1 \vee S^2 \leftarrow \dots $$ of projections. Both are systems of objects of finite type, and both the homotopy limit and homotopy colimit are also of finite type.
We also have a natural isomorphism $$ Map(E_i \wedge X_j, F) \cong \bigoplus_{k=0}^i \bigoplus_{l = 0}^j Map(S^{i+j}, F). $$
Let $F$ be any spectrum, viewed as a constant proobject. Then we can show that the two spectra you describe are different by calculating homotopy classes of maps out to $F$: $$ \begin{align*} Map_{pro}(\{\hocolim_i(E_i\wedge X_j)\}_j, F) &= \hocolim_j \holim_i Map(E_i \wedge X_j, F)\\ Map_{pro}(\hocolim^{pro}_i\{E_i\wedge X_j\}_j, F) &= \holim_i \hocolim_j Map(E_i \wedge X_j, F) \end{align*} $$ Thus, we have isomorphisms $$ \begin{align*} [\{\hocolim_i(E_i\wedge X_j)\}_j, F]_{pro} &= \bigoplus_l \prod_k \pi_{k+l}(F)\\ [\hocolim^{pro}_i\{E_i\wedge X_j\}_j, F]_{pro} &= \prod_k \bigoplus_l \pi_{k+l}(F) \end{align*} $$ In particular, the natural map from the top to the bottom is not an isomorphism unless the homotopy groups of $F$ are zero above a certain degree.
As you can see, finiteness properties on the $E_i$ or $X_j$ don't buy you what you need. If you are taking a hocolim over a finite diagram instead (pushouts, finite wedges, things built from those by finitely many steps), then the fact that the category of prospectra is stable allows you to move the homotopy colimit inside the inverse system.

$\begingroup$ (There are some applications that localize the category of prospectra to ensure that they have convergent Postnikov towers. In that case there might be some assumptions that might work for you.) $\endgroup$ – Tyler Lawson Apr 10 '17 at 0:47

$\begingroup$ Basically, the situation is that the $E_i$ are cohomologically Bousfield localized with respect to $H\mathbb{Z}/p$, and $X$ is $p$profinite, meaning that it is a proobject in the category of $p$finite spaces. The final weak equivalence that I want may also be after cohomologically Bousfield localizing with respect to $H\mathbb{Z}/p$. $\endgroup$ – user108170 Apr 10 '17 at 10:32

2$\begingroup$ @user108170, unfortunately that's not true either. If you let $M = \Bbb{RP}^2$ viewed as a based space, replace $S^n$ with $S^n \wedge M$ throughout, and take $F = \prod_{i=0}^\infty H\Bbb Z/2$, then you can get a contradiction satisfying those conditions too. $\endgroup$ – Tyler Lawson Apr 10 '17 at 17:26