So I recently found out that the rulers of Ethiopia claim descent from Shlomo. Since Mashiach needs to come from David, if this is true they fulfill that requirement. However, it is unlikely (or at least unconfirmable) whether or not they are Jewish due to the matrilineal nature of transmitting Jewishness. Would they qualify for being the descendent of David without having a line of Jewish ancestors? What about if they convert?
In order to inherit and be part of a tribe a person must be be the child of a Jewish couple (both father and mother Jewish) to begin with. For example the blasphemer in the Bible is the son of a Jewish woman of the tribe of Dan (so he was Jewish) and an Egyptian man. Part of the explanation of the dispute is that he was not considered a member of the tribe of Dan. This is explained by Rashi from the Medrash.
י וַיֵּצֵא, בֶּן-אִשָּׁה יִשְׂרְאֵלִית, וְהוּא בֶּן-אִישׁ מִצְרִי, בְּתוֹךְ בְּנֵי יִשְׂרָאֵל; וַיִּנָּצוּ, בַּמַּחֲנֶה, בֶּן
הַיִּשְׂרְאֵלִית, וְאִישׁ הַיִּשְׂרְאֵלִי. 10 And the son of an Israelitish woman, whose father was an Egyptian, went out among the children of Israel; and the son of the Israelitish woman and a man of Israel strove together in the camp.
יא וַיִּקֹּב בֶּן-הָאִשָּׁה הַיִּשְׂרְאֵלִית אֶת-הַשֵּׁם, וַיְקַלֵּל, וַיָּבִיאוּ אֹתוֹ, אֶל-מֹשֶׁה; וְשֵׁם אִמּוֹ שְׁלֹמִית בַּת-דִּבְרִי, לְמַטֵּה-דָן.
11. And the son of the Israelitish woman blasphemed the Name, and cursed; and they brought him unto Moses. And his mother's name was Shelomith, the daughter of Dibri, of the tribe of Dan
A person who is not Jewish because his mother is not Jewish, cannot be considered a member of his father's tribe. For example, Shmuel hanavi wrote the book of Ruth in order to prove that Ruth was a valid convert. Thus, her son by Boaz was a legitimate member of the tribe of Judah and inherited the status of Boaz as being in the "kingly line". Otherwise, David could not have become king. Thus, a person descended from a Jewish father and a nonJewish mother is never considered part of the Jew's "line". Therefore a person cannot be considered a member of the Davidic line if he is not halachically Jewish (or a convert).
The bible (Deuteronymy 17:15) also forbids a nonJew (or convert) from becoming king because he is not a born Jew.
If a kohen has a child with a non-Jewish woman, none of their children are of the tribe of Levi. If a regular Yisrael has children with a non-Jewish woman, none of their children are members of his tribe or can inherit from him. See also the intermarriage statement in the book of Ezra that the men had to send away the non-Jewish wives and children.