When the Jews were taken into captivity and taken to Babylon, how were the thousands of people fed along the way? Was it a death march or was it an organized affair in which people were allowed to stop at various points along the way and hunt for food?

It was not a single event in which thousands of people moved. Rather, there were several times when participants in uprisings against the Babylonians who had captured Judah were transported from their homeland. There is no direct written record of the events; rather, they are reconstructed by historians relying on references to them in ancient texts, especially the Tanakh (Old Testament) books of II Kings and Jeremiah. While there is no description of the travel either to Babylon or returning to Judah, one can only assume from the fact that the Jewish community thrived in exile that the trip was conducted more as a migration than as a death march.

Historians agree that several deportations took place (each the result of uprisings in Palestine), that not all Jews were forced to leave their homeland, that returning Jews left Babylonia at various times, and that some Jews chose to remain in Babylonia-thus constituting the first of numerous Jewish communities living permanently in the Diaspora.

Many scholars cite 597 BCE as the date of the first deportation, for in that year King Jehoiachin was deposed and apparently sent into exile with his family, his court, and thousands of workers. Others say the first deportation followed the destruction of Jerusalem by Nebuchadrezzar in 586; if so, the Jews were held in Babylonian captivity for 48 years. Among those who accept a tradition (Jeremiah 29:10) that the exile lasted 70 years, some choose the dates 608 to 538, others 586 to about 516 (the year when the rebuilt Temple was dedicated in Jerusalem).

https://www.britannica.com/event/Babylonian-Captivity