With a common word like round and many possible derivations all the usages don't have to be linked.

The OED separates the rounders you've talked about into several sub-definitions and possible relations:
I. 1. One who goes round, in special senses:

c. = ROUNDSMAN 1. Usage:1896 in Eng. Dial. Dict. Roundsman is defined as: 1. A labourer in need of parochial relief, who was sent round from one farmer to another for employment, partly at the expense of the farmer and partly at the cost of the parish. [earliest usage 1790's]

d. N. Amer. One who makes the round of prisons, workhouses, drinking saloons, etc.; a habitual criminal, loafer, or drunkard. Also transf. usage examples: 1854 Congress Globe 33rd Congress 1st Sess. App. 1220/3: I have always found him a very kind and agreeable manwhat the 'rounders' in New York would term a 'glover'. 1879 A. DALY Let. 20 Oct. in J. F. Daly Life A. Daly (1917) xxi. 330: [We] are old 'rounders' and familiar with the voice, gait and peculiarities of most of the actors and actresses on the American stage.: 1884 [see REPEATER 5b]. 1891 Boston (Mass.) Jrnl. 7 July 2/4 The regular rounders who are beginning to receive long sentences under the new drunkenness law. 1894 Outing XXIV. 440/2: A gay young bravo, one of New York's many 'rounders', or all-nighters.

f. U.S. slang. A transient railway worker. usage:1908 Casey Jones (song) in Railroad Man's Mag. May 764/1: Come all you rounders, for I want you to hear The story told of an engineer, Casey Jones was the rounder's name, A heavy right-wheeler of a mighty fame. 1939 F. J. LEE Casey Jones 287: The word 'rounder' as applied to Casey must be taken as a light, affectionate appellation. 1961 Listener 24 Aug. 270/2: His was a six-pipe job whose moans sent every coloured 'rounder' from Chicago to New Orleans into ecstasies.