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This paper gives a general coalgebraic account of the notions of possibly infinite trace and possibly infinite execution in state-based, dynamical systems, by extending the generic theory of finite traces and executions developed by Hasuo and coauthors [I. Hasuo, B. Jacobs, and A. Sokolova. Generic trace semantics via coinduction. Logical Methods in Computer Science, 3:1-36, 2007]. The systems we consider are modelled as coalgebras of endofunctors obtained as the composition of a computational type (e.g. nondeterministic or stochastic) with a general transition type. This generalises existing work by Jacobs [B. Jacobs. Trace semantics for coalgebras. In Proc. CMCS 2004, volume 106 of ENTCS, 2004] that only accounts for a nondeterministic computational type. We subsequently introduce path-based temporal (including fixpoint) logics for coalgebras of such endofunctors, whose semantics is based upon the notion of possibly infinite execution. Our approach instantiates to both nondeterministic and stochastic computations, yielding, in particular, path-based fixpoint logics in the style of CTL* for nondeterministic systems, as well as generalisations of the logic PCTL for probabilistic systems.