We study mixed multiple orthogonal polynomials associated with rank-one matrices of measures and their recurrence relations. On the step line we identify the corresponding banded recurrence matrix through the Gauss-Borel factorization of the moment matrix. Elementary Christoffel transformations lead to a factorization of this recurrence matrix into lower and upper bidiagonal factors. For the mixed Piñeiro system we derive contour-integral and generalized hypergeometric representations and formulas for the recurrence and factorization coefficients.