Novel high-efficiency fast-neutron detectors weresuggested for fan-beam tomography applications. They combine multi-layerpolymer converters in gas medium, coupled to thick gaseous electronmultipliers (THGEM). Neutron-induced scattering on the converter's hydrogennuclei results in gas ionization by the escaping recoil-protons between twosuccessive converters. The electrons drift under the action of a homogeneouselectric field, parallel to the converter-foil surfaces, towards aposition-sensitive THGEM multiplying element.In this work we discuss the results of a systematic study of the electrontransport inside a narrow gap between successive converter foils, whichaffects the performance of the detector, both in terms of detectionefficiency and localization properties. The efficiency of transportingionization electrons was measured along a 0.6 mm wide gas gap in 6 and 10 mmwide polymer converters. Computer simulations provided conceptualunderstanding of the observations. For drift lengths of 6 mm, electrons wereefficiently transported along the narrow gas gap with minimaldiffusion-induced losses; an average collection efficiency of 95% wasachieved for ionization electrons induced by few keV photoelectrons. The 10mm height converter yielded considerably lower efficiency due to electricaland mechanical flaws of the converter foils. The results indicate thatdetection efficiencies of ∼ 7% can be expected for 2.5 MeV neutronswith 300-foils converters, of 6 mm height, 0.4 mm thick foils and 0.6 mm gasgap.