OB stars powering stellar bowshock nebulae (SBNe) have been presumed to have large peculiar velocities. We measured peculiar velocities of SBN central stars to assess their kinematics relative to the general O-star population using Gaia EDR3 data for 267 SBN central stars and a sample of 455 Galactic O stars to derive projected velocities v2D. For a subset of each sample, we obtained new optical spectroscopy to measure radial velocities and identify multiple-star systems. We find a minimum multiplicity fraction of 36%{+/-}6% among SBN central stars, consistent with >28% among runaway Galactic O stars. The large multiplicity fraction among runaways implicates very efficient dynamical ejection rather than binary-supernova origins. The median v2D of SBN central stars is v2D =14.6km/s, larger than the median v2D=11.4km/s for non-bowshock O stars. Central stars of SBNe have a runaway (v2D>25km/s) fraction of 24_-7_^+9^%, consistent with the 22_-3_^+3^% for control-sample O stars. Most (76%) SBNe central stars are not runaways. Our analysis of alignment ({Delta}PA) between the nebular morphological and v2D kinematic position angles reveals two populations: a highly aligned ({sigma}PA=25{deg}) population that includes stars with the largest v2D (31% of the sample) and a random (nonaligned) population (69%). SBNe that lie within or near HII regions comprise a larger fraction of this latter component than SBNe in isolated environments, implicating localized ISM flows as a factor shaping their orientations and morphologies. We outline a new conceptual approach to computing the solar local standard of rest motion, yielding [U{sun},V{sun},W{sun}]=[5.5,7.5,4.5]km/s.