We associate two specific projective systems of probability spaces with any Tsirelson convolution system. If the projective limits of these systems exist, then we call the convolution system convergent and $K$-convergent, respectively. It is shown that convergent convolution systems give rise to continuous products of probability spaces, while $K$-convergent convolution systems lead to flow systems. We investigate the relationship between convergence and $K$-convergence, as well as their connections to two-parameter product systems of Hilbert spaces.