Vertical transport in p-i-n diodes containing InGaP/GaAs multi-quantum-wells (MQW) is investigated using current-voltage measurements under forward bias at different temperatures. At low injection, the experimental data are analyzed through a two diode model, by taking into account the non-complete depletion of the MQW region caused by unintentional p-type doping. The diffusion current results to be dominant at high temperatures, whereas the current due to non-radiative recombination through defects in the space charge region becomes more and more relevant as the temperature is reduced. At temperatures above T = 150 K and at high forward voltages, when the current is limited by the series resistance, the thermionic emission of holes over the InGaP barriers controls the transport through the whole MQW region. At lower temperatures resonant tunneling of holes takes place and a simple picture of the hole quantum levels permits to interpret the main details of the I(V) curves at T = 41 K, by supporting the hypothesis of a uniform electric field in the MQW region up to about 100 kV/cm.