Context. Emission lines that are indicative of active accretion have been observed for a number of low-mass companions (M < 30 M_Jup) to stars. Line variability is ubiquitous on stellar accretors, but it has never been characterized in detail for low-mass companions. Such characterizations can offer insights into the accretion mechanism at play. Aims. We aim to characterize the short-to-long-term H I Paschen β emission line variability of two 10 to 30 M_Jup companions on wide orbits: GQ Lup b and GSC 06214-00210 b. We also aim to clarify their accretion mechanisms. Methods. We used J-band observations at R = 1800–2360, obtained with VLT/SINFONI in 2017, to record time-series investigations of the hours-to-weeks variability of the H I Paschen β emission line (1.282 µm). Contrary to H_∝, it should be less affected by chromospheric activity contamination. The photospheric emission was analyzed at each epoch and removed with the ForMoSA forward-modeling tool, using new grids of ATMO models exploring different C/O and [M/H] values. The time series of line profiles and intensities were compared to those of more massive accretors and to predictions from the latest magnetospheric accretion and shock models. To complement these results, we also re-investigated archival spectroscopic observations at near-infrared wavelengths of each target to increase the time frame up to a decade and to build a more comprehensive understanding of the variability processes at play. Results. For GQ Lup b, we find line variability on timescales of several months to decades, whereas it is within the acceptable noise levels on shorter timescales. For GSC 06214-00210 b, we find line variability on timescales of tens of minutes all the way up to a decade. The line profiles of GSC 06214-00210 b are partially resolved in at least one epoch. Both objects show H I Paschen β flux variability that is moderate (<50%), on timescales that are below their rotation period, and that is more significant on longer timescales (up to ~1000% on decade-long timescales). This behavior resembles that of classical T Tauri stars. The line profiles of GQ Lup b are blue-shifted and can only be reproduced by magnetospheric accretion models, while those of GSC 06214-00210 b are fairly well reproduced by both magnetospheric accretion and shock models, except for one epoch for which the shock model is highly favored. The companions have C/O values broadly consistent with solar values. Conclusions. While magnetospheric accretion is favored for GQ Lup b, higher resolution (R > 10 000) observations are required to disentangle the two (non-exclusive) emitting mechanisms. The similar variability behavior observed in these low mass companions and in classical T Tauri stars may support similar accretion mechanisms. The high amplitude of variability on timescales of over a month and longer that is found for both objects could be key to explaining the low yield of H_∝ imaging campaigns.