It's difficult to tie J1 to any specific migration because most studies which have been done do not divide it into subclades. This study attempted to provide some better resolution for J1 in Italy, but the total number of samples for each area is very small. Plus, they only broadly divided it into M267 and the "Arabic" J1e.
These are the different areas sampled in the study:
http://r1b.org/wp-content/uploads/2013/06/U152_Only_Map.png
For what it's worth, these are the results, with M267 percentages first and then those for J1e, now known as J1c3.
I Piemonte/E.Lombardia/W.Liguria: 1.2/.6
II N.E.Italy: 1.4/0
III Bologna: 0/6.9
IV E.Liguria/Toscana: 1.6/1.6
V C.Italy: 2.6/1.3
VI Mainland S.Italy: 2.5/1.5
VII Sicily: 2.1/4.3
VIII Sardinia: 0/2.4
Now the M267 could definitely be better refined into subclades, but I don't think I'd be too out of line to say that perhaps this group reflects the older, more "Neolithic" like lineages, or at least the northern Near East lineages. (I wish that Boattini had dated the two clusters the way that they did the other y lineages, but there was probably too little J1 to do a decent job.)
In looking for a pattern in those results, all I can see is that northern Italy runs from 0 to a range from 1.2 to 1.6%. Once you get to Central Italy you go up about 1% to 2.1 to 2.6%.
In terms of "J1e" I don't see much of a pattern at all to be honest. Most of Italy is in a range from 0 to .6, 1.3, 1.5, 1.6, which are really distinctions without a difference. The exceptions are a slightly higher 2.4 in Sardinia, 4.3 in Sicily, and then a bizarre 6.9 in Bologna.
I don't know whether J1e was in a geographical position at the right time to have participated in the Neolithic migrations, because my understanding is that their major expansion took place during the Bronze Age, but I'm not an expert on J1e so someone correct me if I'm wrong. (There is also the conclusion in the 2009 Chiaroni et al paper that "J1c3 spread with pastoral nomads who would migrate based on rainfall patterns from the
Zagros and
Taurus mountains to the
Levant, with the first such migrations occurring during the neolithic period." I'm not quite sure if that means they might have formed part of the Neolithic migrations to Europe or not. )
Turning to the Muslim invasions in 900 AD, they never got anywhere near Bologna so I think this is some sort of founder effect that arrived under who knows what circumstances, or it's just a function of the small sample. (As Maciamo points out in the general Haplogroup J1 section, the distribution of J1 is indeed spotty in Europe, so it's difficult to draw conclusions) The Sicilian numbers may not be representative either, but even if they are, the absolute level is much smaller than many people might have predicted. Even if we combine these figures with the similarly low E-M81 "Berber" clade numbers, and given that this was a male mediated elite invasion, the total North African/Middle Eastern footprint from this particular event is smaller than I had thought, and so those reports of Frederick II engaging in rather brutal ethnic cleansing seem to be borne out.