Since this is not exactly my area of expertise, I will quote Rex Wallace (Wallace 2011).
He argues that the earliest Latin inscriptions were written from right to left and from left to right (p. 22). He mentions three examples of right-to-left inscriptions in Latin: the Vetusia inscription (ET La 2.1) and the Fibula Praenestina (CIL I².3). Not everyone agrees whether the language of the Vetusia inscription should be considered Latin, though, and the authenticity of the Fibula Praenestina gets questioned from time to time (the artifact, not the text).
Note: Michael Weiss writes, in the new, second edition of his Outline of the Historical and Comparative Grammar of Latin (2020),
"New scientific investigation with a scanning electron microscope by Edilberto Formigli and Daniela Ferro (2014) has now shown beyond a reasonable doubt that both the fibula and its inscription are ancient" (Weiss 2020: 24, ft. 2).
The third example is the bronze Fucine Lake inscription (CIL I².5), which was found in 1877 but is lost now. Clackson and Horrocks 2007/2011 give its text in a new version proposed in Crawford 2006.
This is what it looks like in CIL (de visu, I took the picture myself):
Most importantly, Wallace observes that
"By the middle of the republican period, however, writers had settled on left-to-right direction as the norm, and almost all Latin inscriptions after this date are written in this manner. Inscriptions in other formats, such as boustrophendon, appear rather infrequently" (Wallace 2011: 22).
Also note that there is a difference between boustrophedon (the direction of writing alternates) and sinistroverse (the direction of writing is right-to-left), e.g. a bronze tablet from Falerii Novi (CIL I² 365); please follow the link to see it, since it is copyrighted. Here's the glossed text, from Clackson and Horrocks:
This is what it looks like in CIL (de visu, I took the picture myself):
Clackson and Horrocks 2007 argue it is written in Latin and not Faliscan (contrary to what Getty Images says). As they put it, it is written in the Faliscan manner (sinistroverse), with Faliscan letter forms and mainly Faliscan orthography (see Clackson and Horrock 2007: 119-120 for further detail).
John Bodel (Bodel 2012) observes that by the end of the fifth BCE Latin texts "regularly ran left to right" (p. 82). He discusses two theories, neither of which, he argues, can be proven correct at this point.
According to the first theory, "[w]hen inscribed writing began to
be used to label larger monuments, however, the fixed position of (generally)
rectilinear objects necessitated accommodating a more limited range
of viewing perspectives" (p. 83).
The second theory holds that Roman script is dextroverse because of the desire to make it look as much different as possible from sinistroverse ductus in Etruscan, due to rising political and cultural ambitions of Rome.
References
Bodel, John. 2012. Paragrams, punctuation, and system in ancient Roman script. In S. Houston (ed.), The Shape of script: How and why writing systems change, pp. 63-90. Santa Fe, NM.
Clackson, James, and Geoffrey Horrocks. 2007. The Blackwell history of the Latin language.
Wallace, Rex. 2011. “The Latin Alphabet and Orthography.” In James Clackson (ed.), A Companion to the Latin Language, pp. 9–28. London/New York: Wiley-Blackwell.
