Some class of locally solid topologies (called uniformly $\mu$-continuous) on Köthe-Bochner spaces that are continuous with respect to some natural two-norm convergence are introduced and studied. A characterization of uniformly $\mu$-continuous topologies in terms of some family of pseudonorms is given. The finest uniformly $\mu$-continuous topology ${𝒯}_I^{\varphi }\left(X\right)$ on the Orlicz-Bochner space $L^{\varphi }\left(X\right)$ is a generalized mixed topology in the sense of P. Turpin (see [11, Chapter I]).