The Byzantines brought Albanians to the current area they have populated only as mercenary troops (from Sicily where the Arabs had taken them), that is the reason they use the double headed eagle on a red background as their nations symbol, because that was the exact symbol the Byzantine army used.
In his book, "The Illyrians", John Wilkes gives us the info about the "movement":
With the coming of the Arabs, they converted the Old Albanians (from Caucasus) in the 8th century to Islam. But meanwhile, at the time, the Arabs were waging campaigns in Sicily, dividing it into two parts, (hence there was the Kingdom of the two Sicilies). In order to populate their part of Sicily, the Arabs brought with them Old Albanians from the Caucasus. To this day, their descendants live in Sicily.
Then in 1042, the Byzantine Empire attacked the young Serbian state after having defeated the Arabs in Sicily and having brought the Sicilian Albanians under their command and Christianizing them. The leader of the Byzantines who led the Albanians was named Georgios Maniakos. Maniakos brought Albanian mercenaries from Sicily to fight the Serbs and they settled in two waves in modern day Albania, first the mercenaries came, and then came the women and children. After the defeat of Maniakos, the Byzantines would not let the Albanians return, thus the Albanians requested that the Serbs let them stay on the land. They settled under mount Raban and the city of Berat and from this, the Serbs called them "Rabanasi" or "Arbanasi". The city of Berat was known as Belgrad also, before the Albanians came to settle there. They mostly tended sheep and cattle and lent themselves out to Serbian nobles as brave soldiers.
One of the GREATEST PROOFS that Albanians do come from the Caucasus and that THEY ARE NOT the descendants of the Ancient Illyrians is the Turkish name for the Albanians. "Arnauti", which means "those who have not returned" in Arabic, for the Turks were aware of the origins of the Albanians. And they truly did not return, they stayed in Serbian and Byzantine lands.
And T.J.Winnifrith seems to support this very same theory, we can find in his book: "A History of Northern Epirus/Southern Albania"
The revolts, in 1043, involves Albanians appearing as part of the army of the Byzantine general Maniakos, who, angry at his treatment by the emperor, set out from southern Italy, where he had conducted a brilliant but brutal campaign on behalf of the empire to seize power for himself. He marched the route of the Via Egnatia with an army of Latins and Albanians, carrying all before him meeting an ultimately death near Ostrovo in Greece. The Albanians mentioned by the historian Attaileiates would appear to be part of the Byzantine army, and there is a previous mention of Albanian troops serving in Italy under a general less competent that Maniaces.
From :
Moses Kalankaytuk «History of Aghvan» (Albania) book III, chap. V / also: «Sources on the history of Azerbaijan» (Azerbaycan tarixi uzre qaynaqlar) Baku 1989 p. 53 (in Azeri);
and
«Sources on the history of Azerbaijan» pp. 56-57 / also: Z. M. Baniatov «Review of the sources on the history of Azerbaijan. Arabic sources» (Obzor istochnikov po istorii Azerbaydzhana. Istochniki arabskiye) Baku 1964 p. 5 (in Russian)
we find that :
"By 705 the Arabs completed the invasion of Albania and the country became to be ruled by the Arab feudal lords- emirs. The Albanian church under the pressure from the Arabs and with the direct assistance of the Armenian church had to renounce Dyophisitism: Arabs couldn't tolerate the ideological unity of Albania and Byzantium."
So the conversion and movement of the population can be seen as an actual fact.
Then again we can find more than a few common place-names between the two areas in question:
In the republic of Georgia, in the Caucasus, on the terriory of the former Caucasian Albania, from where the Avar Khanate once had its capital, there is a village named: "Arnauti".
This is the name by which Serbs, Greeks, Turks, FYR-Macedonians and Montenegrin Serbs refer to Albanians in their respective langauges.
There is also village in Georgia named "Bushati", which is the name of an Albanian tribe ("fis") around Lake Skadar. There are three villages named: "Geguti", "Gegeni" and "Gegi". "Ghegheni" is the name designated to Albanians who live north of the Shkumbi River in Albania proper.
And more than enough Albanian - Chechen words to come to the same conclusion :
CHECHENIA=ICHQERIA
ALBANIA=SHQIPTERIA
Chechen=aakharkho,Albanian=katundar,English=peasan t
Chechen=alsamoo,Albanian=me shume,English=more
Chechen=aagan,Albanian=eker,English=wild
Chechen=aara dalan,Albanian=jashte dal,English=get out
Chechen=aaradaqqa,Albanian=terhoqa,English=withdra w
Chechen=aaradovlilla,Albanian=rrugedalje,English=e xit
Chechen=aare,Albanian=rrafsh,English=plain
Chechen=arzha,Albanian=zeze,English=black
Chechen=aaz,Albanian=ze,English=voice
Chechen=baarz,Albanian=varr,English=grave
Chechen=banka,Albanian=burre,English=man
Chechen=baar,Albanian=arre,English=nut
Chechen=bashkhan,Albanian=shkelqyer,English=excell ent
Chechen=bekhka,Albanian=borxh,English=debt,obligat ion
Chechen=bil ma,Albanian=fal me,English=im sorry
Chechen=besan,Albanian=zbehte,English=pale
Chechen=buha,Albanian=buf,English=owl
Chechen=cham,Albanian=shijshem,English=tasty
Chechen=yaalla,waala,Albanian=eja,English=come here
Chechen=chu,Albanian=hyj,English=get in
Chechen=daago,Albanian=djeg,English=burn
Chechen=dahiita,Albanian=dergoj,English=send
Chechen=dehndi,Albanian=gjedhe,English=cattle
Chechen=dain,Albanian=drite,English=light
Chechen=daakhkan,Albanian=gjendem,English=located
Chechen=delqa,Albanian=dreke,English=lunch
Chechen=dowgha,Albanian=djeges,English=hot
Chechen=duq,Albanian=aq,English=so many
Chechen=dyelkha,Albanian=kerkoj,English=to cry
Chechen=eskar,Albanian=ushtri,English=army
Chechen=ghaighanii,Albanian=hidheroj,English=make sad
Chechen=ghaala,Albanian=kala,English=castle
Chechen=gaalat,Albanian=gabim,English=mistake
Chechen=ghishto,Albanian=ngrehine,English=building
Chechen=gharlima,Albanian=ngrirje,English=freezing
Chechen=goola,Albanian=gju,English=knee
Chechen=hakkha,Albanian=terheq,English=draw
Chechen=hoqa,Albanian=kete,English=this
Chechen=hostam,Albanian=gozhde,English=nail
Chechen=khalkhar,Albanian=kercej,English=dance
Chechen=khan,Albanian=kohe,English=time
Chechen=khasbesh,Albanian=kopesht,English=garden
Chechen=keeda,Albanian=qetem,English=cut
Chechen=khena,Albanian=kohe,English=weather
Chechen=khila,Albanian=qene,English=been
Chechen=khilam,Albanian=kam,English=have
Chechen=kho,Albanian=koqeve,English=egg
Chechen=kog,Albanian=kembe,English=leg,foot
Chechen=Iighana,Albanian=Inatosur,English=Angry
Chechen=nana,Albanian=nene,English=mother
Chechen=Ysh,Albanian=Ishin,english=they were
Chechen=shu,Albanian=ju,English=you
I've read another "theory that suggests that :
"The Albanians call themselves "Shqip-tari". This name is not Indo-European in origin and contains in it the Ural-Altaic suffix "ar" or "tar". Much like: "Khaz-AR", "Av-AR", "Magy-AR", "Bulg-AR", "Hung-AR", "Ta-TAR" - "Ship-TAR". Taken together with the Shqiptar-Albanian toponyms on the territory of the former Caucasian Albania, are concidered to be enough proof to support this theory on the etymology of "Shqipatr". "
As for the IE dialect why do the use a numeric system that can be connected to the pre-IE.
Some search will show that the IE has a neat decimal counting system, while the Albanian and French languages show traces of a pre-IE, Old European counting system with base twenty, e.g. in French, 76 is soixante-seize, “60 + 16†(but in Belgian French, septante-six, “70 + 6â€, the normal Romance form), or 80 is quatre-vingts, “4 X 20â€. To be more precise: the analysis of 76 into 70 + 6 (as opposed to 60 + 16) is IE.