Process algebras such as CSP or the Pi-calculus are theories to reason about concurrent software. The Pi-calculus also introduces channel passing to address specific issues in mobility. Despite their similarity, the languages expose salient divergences at the formal level. CSP is built upon trace semantics while labeled transition systems and bisimulation are the privileged tools to discuss the Pi-calculus semantics. In this paper, we try to bring closer both approaches at the theoretical level by showing that proper trace semantics can be built upon the Pi-calculus. Moreover, by introducing locations, we obtain the same discriminative power for both the trace and bisimulation equivalences, in the particular case of early semantics. In a second part, we propose to develop the semantics of a slightly modified language directly in terms of traces. This language retains the full expressive power of the Pi-calculus and most notably supports channel passing. Interestingly, the resulting equivalence, obtained from late semantics, exhibits a nice congruence property over process expressions.