Wet soap foam spontaneously imbibes liquid oil without defoaming when it is brought into contact. The kinetics behind this recently observed phenomenon was studied experimentally, with focus on the origin of the suction force and on the oil front dynamics. Using an aqueous foam with an air volume fraction slightly greater than the critical value phi(c), we show that the pumping pressure of oil and/or miscible liquid into the wet foam is attributed to the interfacial distortion of the bubble surfaces. Two distinct regimes along time t were observed in the oil imbibition dynamics. The proceeding oil front evolves with t(1/2) dependency in the early imbibition time in accordance with the classical theory of penetration of a porous medium, whereas it departs into t(1/3) at late imbibition time. The latter process is attributed to the elongation of an oil branch trapped inside the foam when pumping of the exterior oil has ceased.