We present a study of the radial distribution of dust species in young brown dwarf discs. Our work is based on a compositional analysis of the 10 and 20 μm silicate emission features for brown dwarfs in the Taurus–Auriga star-forming region. A fundamental finding of our work is that brown dwarfs exhibit stronger signs of dust processing in the cold component of the disc, compared to the higher mass T Tauri stars in Taurus. For nearly all of our targets, we find a flat disc structure, which is consistent with the stronger signs of dust processing observed in these discs. For the case of one brown dwarf, 2M04230607, we find the forsterite mass fraction to be a factor of ∼3 higher in the outer disc compared to the inner disc region. Simple large-scale radial mixing cannot account for this gradient in the dust chemical composition, and some local crystalline formation mechanism may be effective in this disc. The relatively high abundance of crystalline silicates in the outer cold regions of brown dwarf discs provides an interesting analogy to comets. In this context, we have discussed the applicability of the various mechanisms that have been proposed for comets on the formation and the outward transport of high-temperature material. We also present Chandra X-ray observations for two Taurus brown dwarfs, 2M04414825 and CFHT-BD-Tau 9. We find 2M04414825, which has an ∼12 per cent crystalline mass fraction, to be more than an order of magnitude brighter in X-ray than CFHT-BD-Tau 9, which has an ∼35 per cent crystalline mass fraction. Combining with previous X-ray data, we find the inner disc crystalline mass fractions to be anti-correlated with the X-ray strength.