Context. In March 2018, the detection by VERITAS of very-high-energy emission (VHE > 100 GeV) from 3C 264 was reported. This is the sixth, and second most distant, radio galaxy ever detected in the TeV regime. Aims. In this article we present a radio and X-ray analysis of the jet in 3C 264. We determine the main physical parameters of the parsec-scale flow and explore the implications of the inferred kinematic structure for radiative models of this γ-ray emitting jet. Methods. The radio data set is comprised of VLBI observations at 15 GHz from the MOJAVE program, and covers a time period of about two years. Through a segmented wavelet decomposition method (WISE code), we estimated the apparent displacement of individual plasma features; we then performed a pixel-based analysis of the stacked image to determine the jet shape. The X-ray data set includes all available observations from the Chandra, XMM, and Swift satellites, and is used, together with archival data in the other bands, to build the spectral energy distribution (SED). Results. Proper motion is mostly detected along the edges of the flow, which appears strongly limb brightened. The apparent speeds increase as a function of distance from the core up to a maximum of ∼11.5 c. This constrains the jet viewing angle to assume relatively small values (θ ≲ 10°). In the acceleration region, extending up to a de-projected distance of ∼4.8 × 10⁴ Schwarzschild radii (∼11 pc), the jet is collimating (r ∝ z^{0.40 ± 0.04}), as predicted for a magnetically-driven plasma flow. By assuming that the core region is indeed magnetically dominated (U_B/U_e > 1), the SED and the jet power can be well reproduced in the framework of leptonic models, provided that the high-energy component is associated to a second emitting region. The possibility that this region is located at the end of the acceleration zone, either in the jet layer or in the spine, is explored in the modeling.