We present a comprehensive study of heavy-hadron production, including $D$ and $B$ mesons, heavy baryons, and the $B_c$ meson. Our calculations are based on the $k_t$-factorization, implemented
within a variable-flavor-number scheme, as initiated in Ref. [1]. The present work completes the initial study by including the scale dependence of fragmentation functions, with a special emphasis on the gluon-to-heavy-hadron contribution.