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We prove functional limit theorems for dynamical systems in the presence of clusters of large values which, when summed and suitably normalised, get collapsed in a jump of the limiting process observed at the same time point. To keep track of the clustering information, which gets lost in the usual Skorohod topologies in the space of càdlàg functions, we introduce a new space which generalises the already more general spaces introduced by Whitt. Our main applications are to hyperbolic and non-uniformly expanding dynamical systems with heavy-tailed observable functions maximised at dynamically linked maximal sets (such as periodic points). We also study limits of extremal processes and record times point processes for observables not necessarily heavy tailed. The applications studied include hyperbolic systems such as Anosov diffeomorphisms, but also non-uniformly expanding maps such as maps with critical points of Benedicks-Carleson type or indifferent fixed points such as Pomeau-Manneville or Liverani-Saussol-Vaienti maps. The main tool is a limit theorem for point processes with decorations derived from a bi-infinite sequence called the transformed anchored tail process.