After Clovis, the Frankish Kingdom was divided, so you would think that whichever Frankish king was in charge of the Rhone valley would be too preoccupied with fighting his brothers in other parts of Francia to invade Italy. Yet, while the Gothic wars were raging, the Franks sent large armies to Italy on multiple occasions. They weren't even allying with the Goths or the Romans, they were just there to conquer and fight and plunder both sides. What was the situation in Francia that allowed them to pull that off?
(Partly taken from this earlier answer)
Although it was traditionally attributed to a mix of Germanic tradition
and a patrimonialisation of the state, current interpretation rather leans towards a pragmatic succession plan. Indeed, such territorial split isn't really observable in late ancient Barbarian polities before the death of Clovis when the warlord's sons were trusted with different regions, in a trend that will continue even after the Merovingian's fall, but without subsequent divisions of the allotted parts (which were rather attributed to descendent or collateral relatives with regular returns to political unity). These divisions weren't seen as the break-away of the regnum (all of the kings being "King of the Franks" or "King of/in [City]") but rather, in a strikingly reminiscent manner to the Roman splits of the late empire in two or more parts, and as for Romans, these divisions had a double objective of preventing disruptive civil wars and succession conflicts, but as well to provide each sub-kings with the means managing the post-imperial state in Gaul and especially a "frontline" to defend and expand.
For the early VIth century, these splits (from Clovis' succession in 511) weren't all together stabilized as they were in three sub-kingdoms (namely Neustria, Austrasia and Burgondia) but more as had oc entities, where kings ruled from a chief city over their parts (Aquitaine being further split itself while Paris being more or less held in indivision).
Not that the political succession was any less prone to dynastic intrigue and conflict than in the Roman Empire (arguably without anti-dynastic usurpations as it happened in Gothic Spain or Lombard Italy, even if Theuderic had to deal with the heirs of the kings of Cologne) : to only mention those issued from Clovis' succession in 511, Theuderic attempted to murder the young Clothar, Clodomir's sons were murdered in 532 by their uncles that took over their brother's part, Theuderic murdered his relative Sigvald to prevent yet another attempt at local usurpation, Clothar overplayed his hand in 534 trying to takeover his dying brother's part and almost got himself defeated doing so. You can notice, however, these are more opportunistic attempts over vulnerable or incapable rulers or possible challengers than everyone fighting anyone in a perfectly anarchic conflict (even the the later 'royal faida', a civil war amongst the successors of Clothar that diminished the kingdom's projection and importance was more akin to a protracted conflict that that).
But while each of Clovis' son had their own interests and policies, the expectation of collegiality seem to have rather worked out up to a point and the expansion of the kingdom was made at the expense of the neighbouring peoples fighting not against one king but usually an alliance of them : Burgundians were eventually subdued by their coalition and their kingdom split between them, Theuderic got support from Clothar defeating and subduing Thuringians, and a similar coalition attempted to takeover the northern-eastern part of the Visigothic kingdom in the 530's. Overall, the sons of Clovis more or less doubled the size of their father's realm while turning it from a more or less set of a warlord's conquest into a (if relatively weak) post-imperial state.
An important factor in their military successes was the death of Theoderic and his grandson Athalaric and trough the troubles of succession in Ostrogothic Italy, peoples attacked by Franks being deprived of Ostrogothic support (as Theoderic gave to Visigoths against Clovis or Burgundians against his sons in exchange of their renouncement of the Provençal region) and thus all the more vulnerable and more prone to see calls Franks to intervene into their own matters. Worse, as Eastern Romans invaded Italy, Franks were in a perfect position to negotiate their neutrality or their alliance in a bidding war between Eastern Romans and Ostrogoths between 536 and 538, each side paying them and Ostrogoths even giving back Provence they felt they couldn't hold against a Frankish attack. While not exactly in the most friendly of terms, each king trying to steal for himself the various subsides received so far, they seem to have maintained a common front dealing with either side.
However, Frankish involvement in Italy from this point onward was exclusively the work of the kings of Metz, Theudebert and Theudebald, his son. First, by sending a contingent of freshly subdued Burgundian : at least attempting a feeble plausible deniability in sending troops that weren't Franks, but possibly also because southern Gallic troops seems to have been much more able in siege warfare, especially as they were sent to help Goths besieging Milan. Already in 511, Theudebert had repelled Danes out of Rhineland using likely non-Frankish troops skilled in fluvial warfare : Frankish domination over Alemanic and Thuringian clients meant kings in the eastern part of the realms had access to not only their tribute but their military service besides the forces and fiscal revenues of their lot in Gaul. Eventually, alongside 'their' Franks, both kings would make a large use of their Germanic clients in Italy.
Giving up pretence of neutrality, Theudebert led a semi-successful raid in northern Italy, being forced to return with bounty due to repent dysentery and failing to convince Ostrogoths to become clients of the Frankish realm, which seem to have been a main focus of Messine involvement in Italy, made even more interesting by Theudebert being the first Frankish and Barbarian king to produce coins at his own effigy in imperial regalia instead of the emperor in Constantinople, possibly hinting at his ambitions in the peninsula but also over his brothers : after all, Theudebert was the first-born and while from a lesser union, the most powerful of Clovis' sons. Although Belisarus sent a very stern letter, it doesn't seem to have done much in curbing the king's intents in the peninsula as he both benefited from his brothers being busy occupying part of Visigothic Spain before being defeated and from Totila's successful ground recovering from Romans in Italy : in the late 440's Theudebert is accounted by Procopios having control over most of modern Piedmont, Lombardia and Venetia.
Details are scarce at best, however, what his control entailed or how it was enforced : Frankish sources don't go much other this, maybe out of familiarity with Messine politics. Arguably, the king's sudden death in 548 and his succession by Theudebald do not seem to have hugely changed things there : as the latter sent Franko-Alemanic armies further in Italy joining with Goths, it's plausible Frankish cpresence was made possible trough troops from Frankish Germania (Alemans, Thuringians, Bavarians, etc.) led by Frankish nobles together with the more or less negotiated support of local elites. But it remains speculative, and the situation on the ground wasn't necessarily much clear eitherto begin with with Gothic nobles holding out in the region still in early 560's without any clue if they acknowledged the Frankish domination or if they did, its nature.
Theudebald himself, said to be as sickly his father was warlike, did not answer for call to help from Goths (neither did his father, hitning there again at some continuity) did allow Goths to seek reinforcement with people from Frankish Germania, coming into the peninsula too late to support Teia but still happy to plunder their way south and for part of them to join with Ostrogothic hold-outs and being utterly defeated by Romans in 554 at the Battle of the Volturno.
Altough the defeat sealed the fate of Frankish ambitions in Italy for two centuries, the reason for a poorly known phasing out of Frankish presence south of the Alps could be more easily related to the situation in Francia : Theudebald died in 555, leaving Chlotar as the main king and critically having to deal not only with a general rebellion in Frankish Germania (Saxons and Thuringians) that is the region where most recruits and armies sent in Italy likely came from, but also in Aquitaine where he had mandated his son Chramm to manage the parts he took from the succession of Theudebald.
The reasons for Chramm's revolt could be related to the temptation of autonomism in Aquitaine at least in some Roman families (after the disappearance of the messine line whose kings had matrimonial relations with) against those closer to Chlotar, with the support of his uncle Childebert (wronged in the same succession). Regardleess, Chramm's revolt would last until 560 and a constant issue for Chlotar to deal with even as Childebert died in 558. Altough the political situation in Francia wasn't one of constant anarchy, rebellion and familiar strife would likely be the reason for Chlotar to not continue his nephew and grand-nephew Italian ambitions, not having at disposal as much as German troops and open rebellion in western Gaul, leaving Romans reconquer northernmost Italy between 556 (where it was reconquered as a whole according Marius of Avranches) and the very last Gothic and/or Frankish hold-outs in the early 560's.
Primary sources
Agathias; Histories
Gregory of Tours; History of the Franks
Marius of Avranches; Chronicle
Procopios; History of the Wars
Secondary sources
Charles Mériaux;La Naissance de la France : les royaumes francs (Ve-VIIe siècles);Belin; 2014.
Bernard S. Bachrach; Merovingian Military Organization, 481-751; University of Minnesota Press; 1972
Bruno Dumézil; Servir l'État Barbare dans la Gaule Franque; 2012; Éditions Tallandier