The unitarily invariant probability measures on infinite Hermitian matrices have been classified by Pickrell, and by Olshanski and Vershik. This classification is equivalent to determining the boundary of a certain inhomogeneous Markov chain with given transition probabilities. This formulation of the problem makes sense for general $β$-ensembles when one takes as the transition probabilities the Dixon-Anderson conditional probability distribution. In this paper we determine the boundary of this Markov chain for any $β\in (0,\infty]$, also giving in this way a new proof of the classical $β=2$ case. Finally, as a by-product of our results we obtain alternative proofs of the almost sure convergence of the rescaled Hua-Pickrell and Laguerre $β$-ensembles to the general $β$ Hua-Pickrell and $β$ Bessel point processes respectively.