  Given a non-negative Jacobi matrix describing higher order recurrence
relations for multiple orthogonal polynomials of type~II and corresponding
linear forms of type I, a general strategy for constructing a pair of
stochastic matrices, dual to each other, is provided. The corresponding Markov
chains (or 1D random walks) allow, in one transition, to reach for the N-th
previous states, to remain in the state or reach for the immediately next
state. The dual Markov chains allow, in one transition, to reach for the N-th
next states, to remain in the state or reach for immediately previous state.
The connection between both dual Markov chains is discussed at the light of the
Poincar\'e's theorem on ratio asymptotics for homogeneous linear recurrence
relations and the Christoffel-Darboux formula within the sequence of multiple
orthogonal polynomials and linear forms of type I.
  The Karlin-McGregor representation formula is extended to both dual random
walks, and applied to the discussion of the corresponding generating functions
and first-passage distributions.
  Recurrent or transient character of the Markov chain is discussed. Steady
state and some conjectures on its existence and the relation with mass points
are also given.
  The Jacobi-Pi\~neiro multiple orthogonal polynomials are taken as a case
study of the described results. For the first time in the literature, an
explicit formula for the type~I Jacobi--Pi\~neiro polynomials is determined.
The region of parameters where the Markov chains are recurrent or transient is
given, and it is conjectured that when recurrent, the Markov chains are null
recurrent and, consequently, the expected return times are infinity. Examples
of recurrent and transient Jacobi--Pi\~neiro random walks are constructed
explicitly.
