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Formation of stars is now believed to be tightly linked to the dynamical evolution of interstellar filaments in which they form. In this paper we analyze the density structure and kinematics of a small network of infrared dark filaments, SDC13, observed in both dust continuum and molecular line emission with the IRAM 30m telescope. These observations reveal the presence of 18 compact sources amongst which the two most massive, MM1 and MM2, are located at the intersection point of the parsec-long filaments. The dense gas velocity and velocity dispersion observed along these filaments show smooth, strongly correlated, gradients. We discuss the origin of the SDC13 velocity field in the context of filament longitudinal collapse. We show that the collapse timescale of the SDC13 filaments (from 1 Myr to 4 Myr depending on the model parameters) is consistent with the presence of Class I sources in them, and argue that, on top of bringing more material to the centre of the system, collapse could generate additional kinematic support against local fragmentation, helping the formation of starless super-Jeans cores. [ABSTRACT FROM AUTHOR]