The earliest phases of clustered star formation and the origin of the stellar initial mass function (IMF) are currently much debated. In order to constrain the origin of the IMF, we investigated the internal and relative motions of starless condensations and protostars previously detected by us in the dust continuum at 1.2mm in the L1688 protocluster of the Ophiuchus molecular cloud complex. The starless condensations have a mass spectrum resembling the IMF and are therefore likely representative of the initial stages of star formation in the protocluster. We carried out detailed molecular line observations, including some N2H+(1-0) mapping, of the Ophiuchus protocluster condensations using the IRAM 30m telescope. We measured subsonic or at most transonic levels of internal turbulence within the condensations, implying virial masses which generally agree within a factor of ~2 with the masses derived from the 1.2mm dust continuum. This supports the notion that most of the L1688 starless condensations are gravitationally bound and prestellar in nature. We measured a global one-dimensional velocity dispersion of less than 0.4km/s between condensations. This small relative velocity dispersion implies that, in general, the condensations do not have time to interact with one another before evolving into pre-main-sequence objects. Our observations support the view that the IMF is partly determined by cloud fragmentation at the prestellar stage. Competitive accretion is unlikely to be the dominant mechanism at the protostellar stage in the Ophiuchus protocluster, but it may possibly govern the growth of starless, self-gravitating condensations initially produced by gravoturbulent fragmentation toward an IMF, Salpeter-like mass spectrum.