  We present basic theoretical constraints on the effects of destruction by
supernovae (SNe) and growth of dust grains in the interstellar medium (ISM) on
the radial distribution of dust in late-type galaxies. The radial gradient of
the dust-to-metals ratio is shown to be essentially flat (zero) if interstellar
dust is not destroyed by SN shock waves and all dust is produced in stars. If
there is net dust destruction by SN shock waves, the dust-to-metals gradient is
flatter than or equal to the metallicity gradient (assuming the gradients have
the same sign). Similarly, if there is net dust growth in the ISM, then the
dust-to-metals gradient is steeper than or equal to the metallicity gradient.
The latter result implies that if dust gradients are steeper than metallicity
gradients, i.e., the dust-to-metals gradients are not flat, then it is unlikely
dust destruction by SN shock waves is an efficient process, while dust growth
must be a significant mechanism for dust production. Moreover, we conclude that
dust-to-metals gradients can be used as a diagnostic for interstellar dust
growth in galaxy discs, where a negative slope indicates dust growth.
