The following theorem is the main result of the paper: Let \((\Omega,{\mathcal E})\) be a measurable space, \(S\) a Souslin space, \(X\) a Hausdorff space with second countable compacts, and \(F:\Omega\times S\to 2^X\) a compact-measurable and closed-valued multifunction. Then the multifunction \(\omega\to\bigcap\{F(\omega, s): s\in S\}\) is universally compact-measurable. The proof is based on the Souslin projection theorem. The author uses some separation ideas developed by A. Spakowski and P. Urbaniec [Rend. Circ. Mat. Palermo, II. Ser. 42, No. 1, 82-92 (1993; Zbl 0793.28011)] and generalizes their result on the measurability of a countable intersection of multifunctions into a countably separated space. In the final part of the paper, he introduces the unique maximal part of a multifunction which is defined on the quotient space given by an equivalence relation. As an application of the main result, he establishes the measurability of this new multifunction in a special case.