This phase I clinical trial studies the side effects and best dose of CD19-specific T-cells in treating patients with lymphoid malignancies that have spread to other places in the body and usually cannot be cured or controlled with treatment. Sometimes researchers change the deoxyribonucleic acid (DNA) (genetic material in cells) of donated T-cells (white blood cells that support the immune system) using a process called "gene transfer." Gene transfer involves drawing blood from the patient, and then separating out the T-cells using a machine. Researchers then perform a gene transfer to change the T-cells' DNA, and then inject the changed T-cells into the body of the patient. Injecting modified T-cells made from the patient may help attack cancer cells in patients with advanced B-cell lymphoma or leukemia.