With a POD of the Italian 1911 invasion of Libya let us say that Britain stands aside and let the Khedive of Egypt permit the passage of Ottoman troops and supplies into Libya across Egypt. What might change from OTL as a result?
The problem is, it's nearly ASB for them to formally allow the Turks to do so. The border actually was fairly porous, as many Muslims tried helping out against the perceived aggression; formally allowing the passage would be seen as an hostile gesture towards Italy and, more importantly, lengthen a War that was already felt as stretching too long. Also of note - the Ottomans did manage to keep Italy from occupying but a tiny coastal strip OTL, so it's not like they needed more troops. They were doomed by a simple lack of true friends - if Austria-Hungary, not exactly an Italy-friendly power, was the one to suggest a peaceful transfer of the province, you can imagine how little the others would care. Italian friendship and stability were both more important...
But let us assume Britain dislikes it, and allows the troops through.
I would then expect a stalemate - the Turks might be able to expel the Italians, but they are hopelessly outclassed on the sea and there might be more serious follow-ups to the occupation of the Dodecanese.
In the end, a negotiated settlement where Italy may not even get Libya, but only stronger, kinda meaningless guarantees on their economical penetration, will ensue; the First Balkan War probably happens similarly to OTL (the Turks are feared more, but can field less defenders as they're warring on two fronts), and Italy probably is less desired as an ally and more distrustful, leading to a later WW1 entry.