Context. High-redshift blazars provide valuable input to studies of the evolution of active galactic nuclei (AGN) jets and provide constraints on cosmological models. Detections at high energies (0.1a< Ea< 100 GeV) of these distant sources are rare, but when they exhibit bright gamma-ray flares, we are able to study them. However, contemporaneous multi-wavelength observations of high-redshift objects (z">"4) during their different periods of activity have not been carried out so far. An excellent opportunity for such a study arose when the blazar TXS 1508+572 (z"="4.31) exhibited a γ-ray flare in 2022 February in the 0.1 300 GeV range with a flux 25 times brighter than the one reported in the in the fourth catalog of the Fermi Large Area Telescope. Aims. Our goal is to monitor the morphological changes, spectral index and opacity variations that could be associated with the preceding γ-ray flare in TXS 1508+572 to find the origin of the high-energy emission in this source. We also plan to compare the source characteristics in the radio band to the blazars in the local Universe (za< 0.1). In addition, we aim to collect quasi-simultaneous data to our multi-wavelength observations of the object, making TXS 1508+572 the first blazar in the early Universe (z">"4) with contemporaneous multi-frequency data available in its high state. Methods. In order to study the parsec-scale structure of the source, we performed three epochs of very-long-baseline interferometry (VLBI) follow-up observations with the Very Long Baseline Array (VLBA) supplemented with the Effelsberg 100-m Telescope at 15, 22, and 43 GHz, which corresponds to 80, 117, and 228 GHz in the rest frame of TXS 1508+572. In addition, one 86 GHz (456 GHz) measurement was performed by the VLBA and the Green Bank Telescope during the first epoch. Results. We present total intensity images from our multi-wavelength VLBI monitoring that reveal significant morphological changes in the parsec-scale structure of TXS 1508+572. The jet proper motion values range from 0.12 mas yr¹ to 0.27 mas yr¹, which corresponds to apparent superluminal motion β_app" 14.3 32.2.c. This is consistent with the high Lorentz factors inferred from the spectral energy distribution (SED) modeling for this source. The core shift measurement reveals no significant impact by the high-energy flare on the distance of the 43-GHz radio core with respect to the central engine, that means this region is probably not affected by e.g., injection of new plasma as seen in other well-studied sources like CTA 102. We determine the average distance from the 43-GHz radio core to the central supermassive black hole to be 46.1 ± 2.3.μas, that corresponds to a projected distance of 0.32 ± 0.02 pc. We estimate the equipartition magnetic field strength 1 pc from the central engine to be on the order of 1.8 G, and the non-equipartition magnetic field strength at the same distance to be about 257 G, the former of which values agrees well with the magnetic field strength measured in low to intermediate redshift AGN. Conclusions. Based on our VLBI analysis, we propose that the γ-ray activity observed in February 2022 is caused by a shock-shock interaction between the jet of TXS 1508+572 and new plasma flowing through this component. Similar phenomena have been observed, for example, in CTA 102 in a shock-shock interaction between a stationary and newly emerging component. In this case, however, the core region was also affected by the flare as the core shift stays consistent throughout the observations.