  The Gaia Data Release 3 (DR3) contains reflectance spectra at visible
wavelengths for 60,518 asteroids over the range between 374-1034 nm,
representing a large sample that is well suited to studies of asteroid
families. We want to assess the potential of Gaia spectra in identifying
asteroid family members. Here, we focus on two L-type families, namely
Tirela/Klumpkea and Watsonia. These families are known for their connection to
Barbarian asteroids, which are potentially abundant in calcium-aluminum rich
inclusions (CAIs). Our method is based (1) on a color taxonomy specifically
built on Gaia data and (2) the similarity of spectra of candidate members with
the template spectrum of a specific family. We identified objects in the halo
of Tirela/Klumpkea, along with possible interlopers. We also found an
independent group of eight asteroids erroneously linked to the family by the
hierarchical clustering method (HCM). Consequently, the knowledge of the size
distribution of the family has been significantly improved, with a more
consistent shape at the larger end. The Watsonia family is a more intricate
case, mainly due to its smaller size and the less marked difference between the
spectral types of the background and of the family members. However, the
spectral selection helps identify objects that were not seen by HCM, including
a cluster separated from the family core by a resonance. For both families, the
V-shape is better defined, leading to a revised age estimation based on the
memberships established mainly from spectral properties. Our work demonstrates
the advantage of combining the classical HCM approach to spectral properties
obtained by Gaia for the study of asteroid families. Future data releases are
expected to further expand the capabilities in this domain
