We use higher-redshift gamma-ray burst (GRB), HII starburst galaxy (HIIG), and quasar angular size (QSO-AS) measurements to constrain six spatially flat and non-flat cosmological models. These three sets of cosmological constraints are mutually consistent. Cosmological constraints from a joint analysis of these data sets are largely consistent with currently-accelerating cosmological expansion as well as with cosmological constraints derived from a combined analysis of Hubble parameter ($H(z)$) and baryon acoustic oscillation (BAO, with Planck-determined baryonic matter density) measurements. A joint analysis of the $H(z)$ + BAO + QSO-AS + HIIG + GRB data provides fairly model-independent determinations of the non-relativistic matter density parameter $Omega_{rm m_0}=0.313pm0.013$ and the Hubble constant $H_0=69.3pm1.2 rm{km s^{-1} Mpc^{-1}}$. These data are consistent with the dark energy being a cosmological constant and with spatial hypersurfaces being flat, but they do not rule out mild dark energy dynamics or a little spatial curvature. We also investigate the effect of including quasar flux measurements in the mix and find no novel conclusions.
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