We present new observational results on the kinematical, morphological, and stellar population properties of a sample of 21 dEs located both in the Virgo Cluster and in the field, which show that 52% of the dEs (1) are rotationally supported, (2) exhibit structural signs of typical rotating systems such as disks, bars, or spiral arms, (3) are younger (~3 Gyr) than non-rotating dEs, and (4) are preferentially located either in the outskirts of Virgo or in the field. This evidence is consistent with the idea that rotationally supported dwarfs are late-type spirals or irregulars that recently entered the cluster and lost their gas through a ram pressure stripping event, quenching their star formation and becoming dEs through passive evolution....