The inwards scattering of planetesimals towards white dwarfs is expected to be a stochastic process with variability on human time-scales. The planetesimals tidally disrupt at the Roche radius, producing dusty debris detectable as excess infrared emission. When sufficiently close to the white dwarf, this debris sublimates and accretes on to the white dwarf and pollutes its atmosphere. Studying this infrared emission around polluted white dwarfs can reveal how this planetary material arrives in their atmospheres. We report a near-infrared monitoring campaign of 34 white dwarfs with infrared excesses with the aim to search for variability in the dust emission. Time series photometry of these white dwarfs from the United Kingdom Infrared Telescope (Wide Field Camera) in the J, H and K bands were obtained over baselines of up to three years. We find no statistically significant variation in the dust emission in all three near-infrared bands. Specifically, we can rule out variability at ~1.3% for the 13 white dwarfs brighter than 16th mag in K band, and at ~10% for the 32 white dwarfs brighter than 18th mag over time-scales of three years. Although to date two white dwarfs, SDSS J095904.69-020047.6 and WD 1226+110, have shown K band variability, in our sample we see no evidence of new K band variability at these levels. One interpretation is that the tidal disruption events which lead to large variabilities are rare, occur on short time-scales, and after a few years the white dwarfs return to being stable in the near-infrared.