Only one of dozens Palestinian clans (khamulas) recognizes its Jewish ancestry. All the others believe that they are descendants of Arabia tribes, or of Salah ad Din settlers (Saladin’s army consisted of Turks, Circassians and Kurds). The rest consider themselves the descendants of 16-20 century settlers from Syria, Lebanon, Transjordan, Egypt, Hejaz, Maghreb, Iraq, Bosnia, etc. Such is the opinion of the PALESTINIANS who remained in Palestine.
Regarding «Christian and Jewish converts to Islam», so the policy of the First caliphs was not to convert Christians and Jews to Islam. The reason was quite practical: the Muslims had exempt from «jizya», the tax paid by Christians and Jews. The Fatimide Caliph al Hakim broke this rule but after his misterious disappearance all his reforms were recalled.
The Crusaders have murdered or expelled most of local Muslims and the Jews, while the Muslims did the same with local Christians after defeating Crusaders.
After the Land emptied of its ancient population, Palestine was settled by newcomers. This way an ancient Palestine population was replaced by new settlers of Muslim, Christian and even Jewish faith. In the Middle Age there ocured demographic overturn in Palestine.
It seems, that Diaspora Jews preserved just ancient genomes, while the ancestors of the Palestinians were of Semitic but not local origin. So the Jews are close but not identical with the Palestinians. Only new research of the genome of pre-exile Palestine population may resolve this scientific and ideological dispute.