  We show that for $\beta \ge 1$ the semigroups of $\beta$-Laguerre and
$\beta$-Jacobi processes of different dimensions are intertwined in analogy to
a similar result for $\beta$-Dyson Brownian motion recently obtained by Ramanan
and Shkolnikov. These intertwining relations generalize to arbitrary $\beta \ge
1$ the ones obtained for $\beta=2$ by the author, O'Connell and Warren between
$h$-transformed Karlin-McGregor semigroups. Moreover they form the key step
towards constructing a multilevel process in a Gelfand-Tsetlin pattern leaving
certain Gibbs measures invariant. Finally as a by product we obtain a relation
between general $\beta$-Jacobi ensembles of different dimensions.
