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Relatively uniformly continuous (ruc) semigroups were recently introduced andstudied by Kandi\'c, Kramar-Fijav\v{z}, and the second-named author, in orderto make the theory of one-parameter operator semigroups available in thesetting of vector lattices, where no norm is present in general.  In this article, we return to the more standard Banach lattice setting -where both ruc semigroups and $C_0$-semigroups are well-defined concepts - andcompare both notions. We show that the ruc semigroups are precisely thosepositive $C_0$-semigroups whose orbits are order bounded for small times.  We then relate this result to three different topics: (i) equality of thespectral and the growth bound for positive $C_0$-semigroups; (ii) a uniformorder boundedness principle which holds for all operator families betweenBanach lattices; and (iii) a description of unbounded order convergence interms of almost everywhere convergence for nets which have an uncountable indexset containing a co-final sequence.

Order boundedness and order continuity properties of positive operator semigroups