  The question of what determines the width of Kuiper belt analogues (exoKuiper
belts) is an open one. If solved, this understanding would provide valuable
insights into the architecture, dynamics, and formation of exoplanetary
systems. Recent observations by ALMA have revealed an apparent paradox in this
field, the presence of radially narrow belts in protoplanetary discs that are
likely the birthplaces of planetesimals, and exoKuiper belts nearly four times
as wide in mature systems. If the parent planetesimals of this type of debris
disc indeed form in these narrow protoplanetary rings via streaming instability
where dust is trapped, we propose that this width dichotomy could naturally
arise if these dust traps form planetesimals whilst migrating radially, e.g. as
caused by a migrating planet. Using the dust evolution software DustPy, we find
that if the initial protoplanetary disc and trap conditions favour planetesimal
formation, dust can still effectively accumulate and form planetesimals as the
trap moves. This leads to a positive correlation between the inward radial
speed and final planetesimal belt width, forming belts up to $\sim$100 au over
10 Myr of evolution. We show that although planetesimal formation is most
efficient in low viscosity ($\alpha = 10^{-4}$) discs with steep dust traps to
trigger the streaming instability, the large widths of most observed
planetesimal belts constrain $\alpha$ to values $\geq4\times 10^{-4}$ at tens
of au, otherwise the traps cannot migrate far enough. Additionally, the large
spread in the widths and radii of exoKuiper belts could be due to different
trap migration speeds (or protoplanetary disc lifetimes) and different starting
locations, respectively. Our work serves as a first step to link exoKuiper
belts and rings in protoplanetary discs.
