  In this note we study a natural measure on plane partitions giving rise to a
certain discrete-time Muttalib-Borodin process (MBP): each time-slice is a
discrete version of a Muttalib-Borodin ensemble (MBE). The process is
determinantal with explicit time-dependent correlation kernel. Moreover, in the
$q \to 1$ limit, it converges to a continuous Jacobi-like MBP with
Muttalib-Borodin marginals supported on the unit interval. This continuous
process is also determinantal with explicit correlation kernel. We study its
hard-edge scaling limit (around 0) to obtain a discrete-time-dependent
generalization of the classical continuous Bessel kernel of random matrix
theory (and, in fact, of the Meijer $G$-kernel as well). We lastly discuss two
related applications: random sampling from such processes, and their
interpretations as models of directed last passage percolation (LPP). In doing
so, we introduce a corner growth model naturally associated to Jacobi
processes, a version of which is the "usual" corner growth of Forrester-Rains
in logarithmic coordinates. The aforementioned hard edge limits for our MBPs
lead to interesting asymptotics for these LPP models. In particular, a special
cases of our LPP asymptotics give rise (via the random matrix Bessel kernel and
following Johansson's lead) to an extremal statistics distribution
interpolating between the Tracy-Widom GUE and the Gumbel distributions.
