When the Pilgrims fled religious persecution in England, they settled in Leiden. They lived and worked in Leiden, and some married there or had children baptized there. They left for America on the Mayflower in 1620, but not after leaving traces in Dutch records.
The Leiden Archives has transcribed several of these records. They used to be available on a separate website, which has been offline for several years. The transcriptions and English abstracts of these records have now been made available on this website.
Go to the Leiden Pilgrim records.
The Pilgrim Fathers. Collection Rijksmuseum (public domain)
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More on Leiden Pilgrims at Familysearch.org:
Leyden documents relating to the Pilgrim fathers, permission to reside at Leyden and betrothal records : together with parallel documents from the Amsterdam archives
Viewable at
https://www.familysearch.org/library/books/records/item/84615-leyden-documents-relating-to-the-pilgrim-fathers-permission-to-reside-at-leyden-and-betrothal-records-together-with-parallel-documents-from-the-amsterdam-archives?viewer=1&offset=0#page=1&viewer=picture&o=info&n=0&q=