  Ratio asymptotics for matrix orthogonal polynomials with recurrence
coefficients $A_n$ and $B_n$ having limits $A$ and $B$ respectively (the matrix
Nevai class) were obtained by Dur\'an. In the present paper we obtain an
alternative description of the limiting ratio. We generalize it to recurrence
coefficients which are asymptotically periodic with higher periodicity, and/or
which are slowly varying in function of a parameter. Under such assumptions, we
also find the limiting zero distribution of the matrix orthogonal polynomials,
generalizing results by Dur\'an-L\'opez-Saff and Dette-Reuther to the
non-Hermitian case. Our proofs are based on "normal family" arguments and on
the solution to a quadratic eigenvalue problem. As an application of our
results we obtain new explicit formulas for the spectral measures of the matrix
Chebyshev polynomials of the first and second kind, and we derive the
asymptotic eigenvalue distribution for a class of random band matrices
generalizing the tridiagonal matrices introduced by Dumitriu-Edelman.
