A novel mechanism to produce a cosmic network of fundamental superstrings based on a time-varying string tension has been recently proposed. It has been found that fundamental superstrings can grow in a kinating background driven by the rolling of the volume modulus of Type IIB string compactifications towards the minimum of its potential. In this talk, I will generalise this analysis using dynamical systems techniques. First, I will analyse the cosmological growth of strings with a field-dependent tension in a Universe filled with a perfect fluid, finding a \emph{growth region} in the phase space of this system. This machinery is then applied to both fundamental superstrings and effective strings obtained from wrapping $p$-branes on $(p-1)$- dimensional cycles. I will show how cosmological growth can be achieved in both cases not only in kinating backgrounds, but also in scaling fixed points. This talk is based on arXiv:2503.11293.