  In contrast to classical strongly continuous semigroups, the study of
bi-continuous semigroups comes with some freedom in the properties of the
associated locally convex topology. This paper aims to give minimal assumptions
in order to recover typical features like tightness and equicontinuity with
respect to the mixed topology as well as to carefully clarify on mutual
relations between previously studied variants of these notions. The abstract
results -- exploiting techniques from topological vector spaces -- are
thoroughly discussed by means of several example classes, such as semigroups on
spaces of bounded continuous functions.
