This phase II Lung-MAP non-Match treatment trial studies how well ramucirumab and pembrolizumab work versus standard of care in treating patients with non-small cell lung cancer that is stage IV or has come back. Immunotherapy with monoclonal antibodies, such as ramucirumab and pembrolizumab, may help the body's immune system attack the cancer, and may interfere with the ability of tumor cells to grow and spread. Drugs used in standard of care chemotherapy for non-small cell lung cancer, such as docetaxel, gemcitabine hydrochloride, and pemetrexed, work in different ways to stop the growth of tumor cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. Giving ramucirumab and pembrolizumab together may work better in treating patients with non-small lung cancer compared to standard of care.

PRIMARY OBJECTIVES: I. To compare overall survival between patients previously treated with platinum-based chemotherapy and immunotherapy for stage IV or recurrent non-small cell lung cancer randomized to ramucirumab and MK-3475 (pembrolizumab) versus standard of care (SoC). SECONDARY OBJECTIVES: I. To compare response rates between the arms, including complete response (CR) and partial response (PR) (confirmed and unconfirmed). II. To compare the disease control rate (CR, PR, confirmed and unconfirmed and stable disease \[SD\]). III. To evaluate the duration of response (DoR) among responders within each arm. IV. To evaluate the frequency and severity of toxicities within each arm. V. To compare investigator assessed-progression-free survival (IA-PFS) between the arms. VI. To evaluate the clinical outcomes (overall survival \[OS\], IA-PFS, response) by randomization stratification factors by comparing outcomes within the ramucirumab and MK-3475 (pembrolizumab) arm, performing a sub-group analysis of the arms, and by evaluating an interaction between the factors and treatment arm. TRANSLATIONAL MEDICINE OBJECTIVES: I. To evaluate if PD-L1 expression levels are associated with clinical outcomes (OS, IA-PFS, and response). II. To evaluate if tumor mutation burden (TMB) as determined by the Foundation Medicine Inc (FMI) Foundation One panel is associated with clinical outcomes. III. To collect, process, and bank cell-free (circulating cell-free deoxyribonucleic acid \[cfDNA\]) at baseline and progression for future development of a proposal to evaluate comprehensive next-generation sequencing of circulating tumor DNA (ctDNA). IV. To establish a tissue/blood repository to pursue future studies. OUTLINE: Patients are randomized to 1 of 2 arms. ARM A: Patients may receive docetaxel intravenously (IV) over 10-30 minutes on day 1, gemcitabine hydrochloride IV over 30 minutes on days 1 and 8, pemetrexed IV over 10 minutes on day 1 (non-squamous NSCLC patients only), or ramucirumab IV over 60 minutes combined with docetaxel IV over 10-30 minutes on day 1. Cycles repeat every 21 days in the absence of disease progression or unacceptable toxicity. ARM B: Patients receive ramucirumab IV over 60 minutes on day 1. Cycles repeat every 21 days in the absence of disease progression or unacceptable toxicity. Patients also receive pembrolizumab IV over 30 minutes on day 1. Treatment repeats every 21 days for up to 35 cycles in the absence of disease progression or unacceptable toxicity. After completion of study treatment (prior to disease progression), patients are followed up every 3 months for the first year, and then every 6 months for up to 3 years from date of randomization. After completion of study treatment (after disease progression), patients are followed up every 6 months for 2 years, then at the end of year 3 from the date of randomization.