its a complex situation.
KRG and Barzani is very much in the Sunni Turkish / Sunni Arab "camp" within this conflict and is only "nominally" anti-isis. He "hosted" the ISIS "families" in Kurdistan, allows them to "trade" via KRG territory as well the famous "ribbentropp molotov" agreement from June 2014 when the "sunni arab" and "kurdish" entities in sync overthrew the "iraqi" government in the northern region and tried overthrowing the government in Baghdad.
The KRG officially doesn't have the "right" to invite any foreign troops (or even issue their own "visa") in without Baghdad's approval... but they've been flouting Iraqi law since 1991 in this regard.
So for the Past 24 years Kurdistan has been able to "invite" foreign forces, issue visas and charge its own taxes and duties on imports as well as run its own borders and exports quite independently from Iraq. At the same time they "participate" in the Baghdad government and make decisions affecting the rest of Iraq as well as take a 12-17% of the non kurdish oil production for their Budget, whilst keeping the revenue "they" generate (from duties, taxes, oil smuggling) for themselves.
When discussing "kurds" its important to note the difference between Barzani/KDP who is an erdogan /sunni arab ally and the other groups who are truly against ISIS and Turkey...
Now there are still Turkish troops in Iraq... and they've actually been there since 1992... but limited to a small strip near the iraqi-turkish border in that airbase... it was used jointly with US forces then.
The recent "open" incursion into Ninawa province on the other hand is "new" precisely because the Turkish forces and Peshmarga have moved into areas which were NOT part of KRG region until the "ribentropp molotov" of June 2014. That's the key difference
this is an "open encroachment" into the non-krg part of Iraq.
I suspect the reasons are many:
-Turks consider Ninawa to be part of Turkey that was "taken away " from them by the British.
-Erdogan supports the local sunni arabs there to form an "ISIS lite" to forestall the storming of Mosul by the approaching "shia" iraqi forces (who are in Shirqat now).
-The "isis" sunni arabs of Iraq look to turkey to "rescue" them from the "shia menace" as well as the yazidis who will displace them once they "retake" the territory... as has happened in all the cities that the "shia forces" have retaken... pro-isis populations have fled towards turkey and the remaining "rump" isis areas to escape from reprisals.
-Erdogan and Co thus realise that if the Iraqi forces end up retaking Mosul... Erdogan and KRG will end taking in 2-3 million refugees from the region who will never be able to return to Mosul... That puts Erdogan in a bind especially as he can't send them all to Europe via the boats...