The 4D low-energy limit of string compactifications is characterised by the ubiquitous presence of string moduli which are new gravitationally coupled scalar fields which develop mass via supersymmetry breaking effects. During inflation it is generically expected that these fields receive large contributions to their mass of order the Hubble constant. They are therefore shifted from their minimum and after the end of inflation start oscillating around it behaving as non-relativistic matter. Given that matter redshifts slower than radiation, the moduli quickly come to dominate the energy density of the universe. Hence, when they decay, they would dilute anything that has been produced before. It is thus crucial to require that the moduli decay before Big-bang nucleosynthesis in order to preserve the successful prediction for the abundances of the light elements. This condition sets a lower bound on their masses of order 50 TeV. This potential problem goes under the name of ``cosmological moduli problem''. In this work, we shall study this problem in promising multi-field inflationary models which naturally emerge in type IIB compactifications. In particular, we will explore if the presence of a large number of spectator fields during inflation can reduce the initial misalignment of the moduli. We shall also explore if the dynamics of the system can forbid a period of moduli domination after the end of inflation.