Tunable transport of tiny objects in fluid systems is demanding in diverse fields of science such as drug delivery, active matter far from equilibrium, and lab-on-a-chip applications. Here, we report the directed motion of colloidal particles and self-assembled colloidal chains in a nematic liquid crystal matrix using electrohydrodynamic convection (EHC) rolls. The asymmetric distortion of the molecular orientation around the particles results - for single particles - in a hopping motion from one EHC roll to the next and - for colloidal chains - in a caterpillar-like motion in the direction perpendicular to the roll axes. We demonstrate the use of colloidal chains as microtraction engines for the transport of various types of microcargo.