L-type ultra-cool dwarfs and brown dwarfs have cloudy atmospheres that could host weather-like phenomena. The detection of photometric or spectral variability would provide insight into unresolved atmospheric heterogeneities, such as holes in a global cloud deck. Indeed, a number of ultra-cool dwarfs have been reported to vary. Additional time-resolved spectral observations of brown dwarfs offer the opportunity for further constraining and characterising atmospheric variability. Aims. It has been proposed that growth of heterogeneities in the global cloud deck may account for the L- to T-type transition when brown dwarf photospheres evolve from cloudy to clear conditions. Such a mechanism is compatible with variability. We searched for variability in the spectra of five L6 to T6 brown dwarfs to test this hypothesis. Methods. We obtained spectroscopic time series using the near-infrared spectrographs ISAAC on VLT-ANTU, over 0.99-1.13µm, and SpeX on the Infrared Telescope Facility for two of our targets in the J, H, and K bands. We searched for statistically variable lines and for a correlation between those. Results. High spectral-frequency variations are seen in some objects, but these detections are marginal and need to be confirmed. We find no evidence of large-amplitude variations in spectral morphology and we place firm upper limits of 2 to 3% on broad-band variability, depending on the targets and wavelengths, on the time scale of a few hours. In contrast to the rest of the sample, the T2 transition brown dwarf SDSS J1254-0122 shows numerous variable features, but a secure variability diagnosis would require further observations. Conclusions. Assuming that any variability arises from the rotation of patterns of large-scale clear and cloudy regions across the surface, we find that the typical physical scale of cloud-cover disruption should be smaller than 5-8% of the disk area for four of our targets, using simplistic heterogeneous atmospheric models. The possible variations seen in SDSS J1254-0122 are not strong enough to allow us to confirm the cloud-breaking hypothesis.