For food and most other life history topics no information is available. It can be assumed that it will have shared many (but not all) habits with its closest relative, the Red-legged Thrush (Turdus plumbeus).
Larsen, N. (2014). Grand Cayman Thrush (Turdus ravidus), version 1.0. In Neotropical Birds Online (T. S. Schulenberg, Editor). Cornell Lab of Ornithology, Ithaca, NY, USA. https://doi.org/10.2173/nb.grcthr1.01