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This work investigates models and numerical strategies for the behavior of a slender object deformed by large external forces in spinning processes. Our main consideration is an incompressible, highly viscous and three-dimensional jet that is described by a one-dimensional model of partial differential-algebraic equations. We aim to provide a robust basis for the simulation of production processes that require transient treatment and prevent any meaningful simplification of the model equations. The spatial domain is considered time-dependent and requires proper handling. For that purpose, a Finite Volume method for an arbitrary space-time domain is proposed. The performance of the model and discrete scheme is validated through numerical convergence order results (in space, time and combined). As examples for industrial applications we consider production processes of insulation with the rotational spinning process and nonwoven materials with the melt-blowing process. Both exhibit large elongations that manifest in strongly varying solution components, possibly causing numerical difficulties. Possibilities with adaptive mesh refinement (in particular r-refinement) are also explored.