Let $X_{\bullet}$ be a simplicial space and denote the (non-fat!) geometric realization by $\lvert X_{\bullet} \rvert$.

Does this geometric realization of simplicial spaces preserve finite limits?

This is well-known to be true for *simplicial sets* instead of simplicial spaces. Moreover, the fat geometric realization preserves finite limits up to homotopy and the fat-free geometric realization preserves pullbacks and products on the nose (see Proposition 8 and 9 here for references to proofs). So there might be some hope.