  We study the finiteness properties of the braided Higman--Thompson group
$bV_{d,r}(H)$ with labels in $H\leq B_d$, and $bF_{d,r}(H)$ and $bT_{d,r}(H)$
with labels in $H\leq PB_d$ where $B_d$ is the braid group with $d$ strings and
$PB_d$ is its pure braid subgroup. We show that for all $d\geq 2$ and $r\geq
1$, the group $bV_{d,r}(H)$ (resp. $bT_{d,r}(H)$ or $bF_{d,r}(H)$) is of type
$F_n$ if and only if $H$ is. Our result in particular confirms a recent
conjecture of Aroca and Cumplido.

Group Theory;Algebraic Topology;K-Theory and Homology