I have read various theories about the origins of the Mongolian script, which I believe is known as the Hudum Mongol Bichig. Some claim it is ultimately descended from Sogdian runes(which means its oldest ancestor is Sumerian cuneiform), but others claim it was derived from the Khitan script(developed by the Khitan people who were genetically related to the Mongols), which was descended from Hanzhi(Chinese writing). What is the evidence though for either of these theories?
There is some confusion caused by the fact that Old Turkic Script is also known as Turkic Runes, which is a metaphoric application of the term applied to Germanic alphabets. Old Turkic Script is a descendant of Sogdian script, as is Old Uyghur Script, and Sogdian traces back to the Phoenician alphabe, which ultimately comes from Egyptian hieroglyphics. It is theorized that Mesopotamian picture-writing is the inspirational source of all western writing systems, but the actual letter shapes of Egyptian vs. Sumerian writing are not the same.
We needn't get highly technical to understand that Mongolian script has no resemblance to Khitan script, which an ordinary person would think was Chinese. The other Khitan script also looks like Chinese (unicode table here). We don't know how to read the script, but it seems to be a case of taking the overall design of Chinese characters and using then in a Uyghur-like fashion (as a phonetic script, not a logographic one). It is thus valid to consider Khitan to be a conceptual amalgam of Chinese and Sogdian writing systems.
The Sogdian alphabet is at least superficially much more similar to Old Uyghur, for example "lamedh" ([l]) is similar in both systems, at least based on the contemporary standardized shapes of letters (one should look at photographs of actually-ancient writing to firm up this conclusion). Old Turkic Script looks less like Mongolian, but also note the variation between the Orkhon and Yenisean Turkic scripts.
Unlike people, writing systems don't have definite parentage. Many writing systems have been developed which used the pick-and-choose method of creation, and they are highly influence by their physical environment (ink on paper? incisions of woody substances? chiseled in stone? pressed in clay?).