We use Keck/DEIMOS spectroscopy to confirm the cluster membership of 16 ultra-diffuse galaxies (UDGs) in the Coma cluster, bringing the total number of spectroscopically con- firmed UDGs to 24. We also identify a new cluster background UDG. In this pilot study of Coma UDGs in velocity phase-space, we find evidence that most present-day Coma UDGs have a recent infall epoch while a few may be ancient infalls. These recent infall UDGs have higher absolute relative line-of-sight velocities, bluer optical colors, and are smaller in size, unlike the ancient infalls. The kinematics of the spectroscopically confirmed Coma UDG sample is similar to that of the cluster late-type galaxy population. Our velocity phase-space analysis suggests that present-day cluster UDGs have a predominantly accretion origin from the field, acquire velocities corresponding to the mass of the cluster at accretion as they are accelerated towards the cluster center, and become redder and bigger as they experience the various physical processes at work within the cluster.