The question is tagged native-americans and alaska though uses the term new-world, which is a purely Eurocentric perspective. Also, it is not immediately clear what "Old World" means in regards to locales other than Europe proper; or if the question is focused on actual Native Americans or individuals and institutions who invaded Turtle Island and now claim the geography as their own by right of conquest. It must also be noted here that Native American prisoners of war (commonly referred to as "slaves") were shipped to Europe from the "New World" prior to the abolition of "slavery" in the "New World"; which is not migration, but rather a tactic of war to seize and control the land of the original people in the "New World"; i.e.g., see Colonists shipped Native Americans abroad as slaves by Gillian Kiley-Brown
While natives had been forced into slavery and servitude as early as 1636, it was not until King Philip’s War that natives were enslaved in large numbers, Fisher writes in the study. The 1675 to 1676 war pitted Native American leader King Philip, also known as Metacom, and his allies against the English colonial settlers.
During the war, New England colonies routinely shipped Native Americans as slaves to Barbados, Bermuda, Jamaica, the Azores, Spain, and Tangier in North Africa, Fisher says.
Have inquired into why people of European descent do not migrate or "caravan" back to Europe en masse. The conclusion that have drawn is that the conditions which precipitated mass departure from Europe between 1500 and 1900 still exist; and people who claim to be of European descent in the "New World" actually have little interest in returning to the lands and culture they claim by virtue of purported ancestral lineage or "origin", as the evidence supports.
People who claim to be "Jew" do more frequently migrate to Isreal which could be considered "Old World", unless, again by the term "Old World" the question refers to only regions of Europe proper, or euphemistically; that is, the question does not present definitive nations or geographic locations specifying precisely where "Old World" supposedly begins or concludes, certainly not from the perspective of individuals who do not self-identify as "European"; or if such notions of "Old World" exist primarily as nostalgia in individuals' minds who claim to be of European descent, as an expression of Eurocentrism, negating the fact that "Old World" could also be applicable to "Africa" or "Australia", et al.; as the designation "Old World" is not commonly found on any map.
In any event, see Aliyah
(US: /ˌæliˈɑː/, UK: /ˌɑː-/; Hebrew: עֲלִיָּה aliyah, "ascent") is the immigration of Jews from the diaspora to the Land of Israel (Eretz Israel in Hebrew). Also defined as "the act of going up"—that is, towards Jerusalem—"making Aliyah" by moving to the Land of Israel is one of the most basic tenets of Zionism.
Aliyah from Latin America
In the 1999–2002 Argentine political and economic crisis that caused a run on the banks, wiped out billions of dollars in deposits and decimated Argentina's middle class, most of the country's estimated 200,000 Jews were directly affected. Some 4,400 chose to start over and move to Israel, where they saw opportunity.
More than 10,000 Argentine Jews immigrated to Israel since 2000, joining the thousands of previous Argentine immigrants already there. The crisis in Argentina also affected its neighbour country Uruguay, from which about half of its 40,000-strong Jewish community left, mainly to Israel, in the same period. During 2002 and 2003 the Jewish Agency for Israel launched an intensive public campaign to promote aliyah from the region, and offered additional economic aid for immigrants from Argentina. Although the economy of Argentina improved, and some who had immigrated to Israel from Argentina moved back following South American country's economic growth from 2003 onwards, Argentine Jews continue to immigrate to Israel, albeit in smaller numbers than before. The Argentine community in Israel is about 50,000-70,000 people, the largest Latin American group in the country.
There has also been immigration from other Latin American countries that have experienced crises, though they have come in smaller numbers and are not eligible for the same economic benefits as immigrants to Israel from Argentina.
In Venezuela, growing antisemitism in the country, including antisemitic violence, caused an increasing number of Jews to move to Israel during the 2000s. For the first time in Venezuelan history, Jews began leaving for Israel in the hundreds. By November 2010, more than half of Venezuela's 20,000-strong Jewish community had left the country.51, 52
Aliyah from North America
More than 200,000 North American immigrants live in Israel. There has been a steady flow of immigration from North America since Israel’s inception in 1948.82, 83
Several thousand American Jews moved to Mandate Palestine before the State of Israel was established. From Israel's establishment in 1948 to the Six-Day War in 1967, aliyah from the United States and Canada was minimal. In 1959, a former President of the Association of Americans and Canadians in Israel estimated that out of the 35,000 American and Canadian Jews who had made aliyah, only 6,000 remained.84
Following the Six-Day War in 1967, and the subsequent euphoria among world Jewry, significant numbers arrived in the late 1960s and 1970s, whereas it had been a mere trickle before. Between 1967 and 1973, 60,000 North American Jews immigrated to Israel. However, many of them later returned to their original countries. An estimated 58% of American Jews who immigrated to Israel between 1961 and 1972 ended up returning to the United States.85, 86
Like Western European immigrants, North Americans tend to immigrate to Israel more for religious, ideological, and political purposes, and not financial or security ones.87 Many immigrants began arriving in Israel after the First and Second Intifada, with a total of 3,052 arriving in 2005 — the highest number since 1983.88
Nefesh B'Nefesh, founded in 2002 by Rabbi Yehoshua Fass and Tony Gelbart, works to encourage Aliyah from North America and the UK by providing financial assistance, employment services and streamlined governmental procedures. Nefesh B’Nefesh works in cooperation with the Jewish Agency and the Israeli Government in increasing the numbers of North American and British immigrants.
Following the Global Financial Crisis in the late 2000s, American Jewish immigration to Israel rose. This wave of immigration was triggered by Israel's lower unemployment rate, combined with financial incentives offered to new Jewish immigrants. In 2009, aliyah was at its highest in 36 years, with 3,324 North American Jews making aliyah.89