The AllWISE processing pipeline has measured motions for all objects detected on WISE images taken between 2010 January and 2011 February. In this paper, we discuss new capabilities made to the software pipeline in order to make motion measurements possible, and we characterize the resulting data products for use by future researchers. Using a stringent set of selection criteria, we find 22,445 objects that have significant AllWISE motions, of which 3,525 have motions that can be independently confirmed from earlier 2MASS images yet lack any published motions in SIMBAD. Another 58 sources lack 2MASS counterparts and are presented as motion candidates only. Limited spectroscopic follow-up of this list has already revealed eight new L subdwarfs. These may provide the first hints of a "subdwarf gap" at mid-L types that would indicate the break between the stellar and substellar populations at low metallicities (i.e., old ages). Another object in the motion list -- WISEA J154045.67-510139.3 -- is a bright (J ~ 9 mag) object of type M6; both the spectrophotometric distance and a crude preliminary parallax place it ~6 pc from the Sun. We also compare our list of motion objects to the recently published list of 762 WISE motion objects from Luhman (2014). While these first large motion studies with WISE data have been very successful in revealing previously overlooked nearby dwarfs, both studies missed objects that the other found, demonstrating that many other nearby objects likely await discovery in the AllWISE data products. ; Comment: Accepted for publication in The Astrophysical Journal. Table 3 is a machine-readable ASCII file