Scaled momentum distributions for the strange hadrons K-S(0) and Lambda/(Lambda) over bar were measured in deep inelastic ep scattering with the ZEUS detector at HERA using an integrated luminosity of 330 pb(-1). The evolution of these distributions with h the photon virtuality, Q(2), was studied in the kinematic region 10 < Q(2) < 40000 CeV2 and 0.001 < x < 0.75, where x is the Bjorken scaling variable. Clear scaling violations are observed. Predictions based on different approaches to fragmentation were compared to the measurements. Leading-logarithm parton-shower Monte Carlo calculations interfaced to the Lund string fragmentation model describe the data reasonably well in the whole range measured. Next-to-leading-order QC{[}) calculations based on fragmentation functions, FFs, extracted front e(+)e(-) data alone, fail to describe the measurements. The calculations based on FFs extracted from a global analysis including e(+)e(-), ep and pp data give an improved description. The measurements presented in this paper have the potential to further constrain the FFs of quarks, anti-quarks and gluons yielding K-S(0) and Lambda/(Lambda) over bar strange hadrons.