We see the R-L2 dominated people to be the leading element in the Tumulus culture invasion of Pannonia and the Balkans and they just crushed into the local populations and caused a massive demographic decline in the region. Before you had large settlements and agrarian structures, many finds, high level of production and interaction, then comes the TC peopel with the Koszider horizon and in some regions no people lived any more, in others just half or a one third of the former population.
And most importantly, you see in the data that J-L283 and E-V13 both suffered equally under this onslaught. But soon after, R-L2 and J-L283 go into synchrony and grow rapidly, for the first time together. So two formerly unrelated and hostile people seem to have shared some kind of network and success story. This doesn't tell us that the TC R-L2 people did subjugate or control the local J-L283 people, but some kind of source - receiver relationship was there. And as you say, the TC influenced cultures in the North Western Balkans being clearly related to the wider TC networks.
So the least we have to recognise is that the J-L283 people got integrated into this wider Middle Danubian TC networks. E-V13, and this is noteworthy, was not. On the contrary, you see that R-L2 and J-L283 expand in very early UF, causing yet another dent for E-V13, but then when E-V13 started to gain control, with Gáva and Channelled Ware in Late Urnfield, with improved weaponry and first iron weapons, both R-L2 and J-L283 go down against E-V13:
Rather it seems that after that, R-L2 and J-L283, after having a shared fate for some generations, depart again. My guess is because the E-V13 dominated late UF, Thraco-Cimmerian horizon and beginning Hallstatt caused a wedge between, final break between the cremating Northern groups and the inhumation burial proper Illyrian groups.
Before I made those graphs with YFull data, I thought J-L283 might have joined the TC ranks early on, but that's definitely not the case. They almost got annihilated by the TC/R-L2 dominated people, but then they somehow-someway seem to have joined their ranks. Whether they kept their language while doing so or adopted the language of the TC people is impossible to know. But especially in and around Slovenia, there is a fluent-smooth and highly mixed borderzone between R-L2 and J-L283 it seems to me.
Quite typically, the other Balkan lineages didn't recover from this like J-L283, and E-V13 did so on its own, so its a really different, quite peculiar pattern. You can also see, that R-L2 didn't do as well after the Cimmerian invasion/Thraco-Cimmerian horizon, the "native" J-L283 and E-V13 take over.