The kiloparsec-scale kinematics and density structure of the circumgalactic medium (CGM) is still poorly constrained observationally, which poses a problem for understanding the role of the baryon cycle in galaxy evolution. Here we present VLT/MUSE integral-field spectroscopy (R approximate to 1800) of four giant gravitational arcs exhibiting W-0 greater than or similar to 0.2 & Aring; C IV absorption at eight intervening redshifts, z(abs) approximate to 2.0-2.5. We detected C IV absorption in a total of 222 adjacent and seeing-uncorrelated sight lines whose spectra sample beams of ("de-lensed") linear size approximate to 1 kpc. Our data show that (1) absorption velocities cluster at all probed transverse scales, Delta r(perpendicular to) approximate to 0-15 kpc, depending on system; (2) the (transverse) velocity dispersion never exceeds the mean (line-of-sight) absorption spread; and (3) the (transverse) velocity autocorrelation function does not resolve kinematic patterns at the above spatial scales, but its velocity projection, xi(arc)(Delta v), exhibits a similar shape to the known two-point correlation function toward quasars, xi(QSO)(Delta v). An empirical kinematic model suggests that these results are a natural consequence of wide-beam observations of an unresolved clumpy medium. Our model recovers both the underlying velocity dispersion of the clumps (70-170 km s(-1)) and the mean number of clumps per unit area (2-13 kpc(-2)). The latter constrains the projected mean inter-clump distance to within approximate to 0.3-0.8 kpc, which we argue is a measure of clump size for a near-unity covering fraction. The model is also able to predict xi(arc)(Delta v) from xi(QSO)(Delta v), suggesting that the strong systems that shape xi(arc)(Delta v) and the line-of-sight velocity components that define xi(QSO)(Delta v) trace the same kinematic population. Consequently, the clumps must possess an internal density structure that generates both weak and strong components. We discuss how our interpretation is consistent with previous observations using background galaxies and multiple quasars as well as its implications for the connection between the small-scale kinematic structure of the CGM and galactic-scale accretion and feedback processes.