Basically some variation of "Jews kill some Chrisitan and use the blood in matzah." What are the origins? How did they survive and what did people think the "motive" was?
So this is called a "blood libel", which is a common antisemitic trope. It refers more broadly to religious claims involving ritual murder by Jews. The blood in matzah one is the best known today, but there were other elements to these slanders. Often the claim would be that the Christian child in question was murdered in a way intended to emulate the crucifixion of Jesus. This connection is important, because Passover occurs at roughly the same time as Easter, so there's a connection between the blood-in-Matzah blood libels and the claim of Jewish deicide (blaming Jews for killing Jesus).
As a consequence, Christian violence against Jews was so common during the Easter season that in many areas it was common for either local officials or the Jewish community to hire guards for the duration of the week, or for the weekend of Easter, to mitigate the threat. But this annual violence was quite common, and often accepted by both the populace and the Christian clergy. It also isn't necessarily directly related to blood libels, but it's some background. Another related thing is host desacration claims, where Jews were accused of somehow capturing Catholic communion, and performing some sort of violent ritual with it that was supposedly a reenactment of Jesus being killed. That's quite similar to the blood libel, because of the same concept that Jews were somehow reenacting killing Jesus in an offensive way (either through murder or through host desecration).
Anyway, back to the main topic. The first blood libel claims are from Western Europe in the 12th century. The first recognizable one is from 1144 in Norwich, England. A boy was found murdered around the time of Easter. Blame for this was fixed to the Jews. While the trope of blood-in-Matzah is not found here, it was suggested in accounts of this that the victim was tortured in such a way as to resemble Jesus' wounds. The victim was seen (though not universally among local Christians) as a martyr, and miracles were ascribed to them. These claims continued in the 12th century, but did not result in actual violence towards Jews in England. The first serious anti-Jewish violence in England occurred for unrelated reasons in 1189-1190, but there were blood-libel-related outbreaks of anti-Jewish violence related to blood libels in the 13th century in England, prior to the expulsion of English Jews in 1290.
Soon after the origin of these claims in England, similar stories proliferated in Europe, often with even less evidence (in 1144 in Norwich, a boy was actually murdered, and a Christian man was in Germany in 1147, unlike 1171 in Blois, which resulted in the execution of 30 Jews). From England these blood libels were first (and most frequently) found in Germany, then in Northern France, and subsequently elsewhere in Europe.
In the 18-19th centuries, as you mention, blood libels continued to be spread, and were a source of violence against Jews in Eastern Europe, most notably in the Kishinev pogrom. The blood-in-matzah, while the most sensational bit, is not a necessary component of blood libels, but it was present in the Kishinev Pogrom's leadup.
As for its ultimate origin, the best guess from the English accounts is that it was progressive rumor-spreading. First a boy was murdered, then a rumor spread that the Jews were responsible, then it was later claimed that the murder was ritual, etc. There are earlier precedents (Josephus refuted a Greek claim that Jews performed human sacrifice of a Greek man in the temple), but it's unlikely that these were related to the outbreak of blood libels later on, which originated in 1144 in Norwich.
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