There are arguments about whether the Tulu script descended from the Malayalam script or whether Malayalam descended from Tulu. I tend to believe the camp which posits that there was a single script, derived from Grantha, which diverged into Tulu and Malayalam at some point.
Unfortunately for Tulu speakers, when German missionaries showed up in the 1800s and printed tracts in the Tulu language, they chose to use the Kannada script instead of the Tulu script. They already had a press for Kannada (and hence already had Kannada metal type), and almost all Tulu speakers also spoke Kannada.
We in the 21st century tend to think of printing as being something cheap and easy to do; it is difficult to remember how expensive and difficult it was to create a new typeface for a new language. As I mentioned in my Khojki post, it took Mr. Laljibhi Devraj a trip to Germany and three months of his time to get metal type cut for Kohjki. Thus, given that type existed for Kannada and it worked well enough for Tulu, it is not surprising that the Tulu script would lose popularity.