We derive density evolution equations for a belief-propagation (BP) based detection algorithm for a vector channel under assumption of random channel parameters and in the large-system limit, considering the conditions of a general information symbol prior probability and a general memoryless channel. From the density evolution equations, it is observed that there exists a bank of independent single-user Gaussian channels which is equivalent to the original vector channel at each stage of the BP-based detection algorithm in the sense that the posterior probabilities of information symbol of a user are equivalent to each other. It means that the decoupling principle holds not only at equilibrium, but also at each stage of the algorithm. We also show a microscopic stability condition for the BP-based detection algorithm, in order to discuss possible relationship with the stability of replica-symmetric solutions against replica symmetry breaking in the replica analysis.