Docetaxel With or Without Bintrafusp Alfa for the Treatment of Advanced Non-small Cell Lung Cancer

Inclusion Criteria: * Age \>= 18 years * Histological confirmation of non-small cell lung cancer (NSCLC) with advanced disease * Prior treatment required: * Anti-PD1/PD-L1 agent in combination with platinum-based chemotherapy * Measurable disease * NOTE: Tumor lesions in a previously irradiated area are not considered measurable disease; Disease that is measurable by physical examination only is not eligible * Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0, 1 or 2 * Hemoglobin \>= 9.0 g/dL (obtained =\< 14 days prior to registration) * Absolute neutrophil count (ANC) \>= 1500/mm\^3 (obtained =\< 14 days prior to registration) * Platelet count \>= 100,000/mm\^3 (obtained =\< 14 days prior to registration) * Total bilirubin =\< upper limit of normal (ULN) (obtained =\< 14 days prior to registration) * Alanine aminotransferase (ALT/serum glutamate pyruvate transaminase \[SGPT\]) and aspartate transaminase (AST/serum glutamic-oxaloacetic transaminase \[SGOT\]) =\< 1.5 x ULN (obtained =\< 14 days prior to registration) * Alkaline phosphatase \<= 2.5 x ULN (obtained =\< 14 days prior to registration) * Prothrombin time (PT)/international normalized ratio (INR)/activated partial thromboplastin time (aPTT) =\< 1.5 x ULN OR if patient is receiving anticoagulant therapy INR or aPTT is within target range of therapy (obtained =\< 14 days prior to registration) * Calculated creatinine clearance \>= 30 ml/min using the Cockcroft-Gault formula (obtained =\< 14 days prior to registration) * Negative pregnancy test done =\< 7 days prior to registration, for persons of childbearing potential only * NOTE: If a urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required * Willing to use birth control as follows: * If able to become pregnant: Willing to use birth control during treatment and for 6 months after last dose of docetaxel and/or bintrafusp alfa, whichever is later * If able to father a child: Willing to use birth control with partners able to become pregnant during treatment and for 3 months after last dose of docetaxel and/or bintrafusp alfa, whichever is later * Provide written informed consent * Willingness to provide mandatory blood specimens for correlative research * Willingness to provide mandatory tissue specimens for correlative research * Willing to return to enrolling institution for follow-up (during the active monitoring phase of the study) * CROSSOVER ELIGIBILITY: Documented disease progression =\< 28 days prior to crossover registration * CROSSOVER ELIGIBILITY: No contraindications to bintrafusp alfa at the time of crossover registration * CROSSOVER ELIGIBILITY: Patient and physician agree to try crossover treatment with bintrafusp alfa * CROSSOVER ELIGIBILITY: Provide written informed consent Exclusion Criteria: * Any of the following because this study involves an investigational agent whose genotoxic, mutagenic and teratogenic effects on the developing fetus and newborn are unknown: * Pregnant persons * Nursing persons * Persons of childbearing potential who are unwilling to employ adequate contraception * Any of the following prior therapies: * Surgery =\< 4 weeks prior to registration * Chemotherapy =\< 4 weeks prior to registration * Received single agent anti-PD1/PD-L1 as first line therapy for metastatic disease * Co-morbid systemic illnesses or other severe concurrent disease which, in the judgment of the investigator, would make the patient inappropriate for entry into this study or interfere significantly with the proper assessment of safety and toxicity of the prescribed regimens * Uncontrolled intercurrent illness including, but not limited to: * Ongoing or active infection * Symptomatic congestive heart failure * Unstable angina pectoris * Cardiac arrhythmia * Or psychiatric illness/social situations that would limit compliance with study requirements * Receiving any other investigational agent which would be considered as a treatment for the primary neoplasm * Other active malignancy =\< 5 years prior to registration * EXCEPTIONS: Non-melanotic skin cancer or carcinoma-in-situ of the cervix * NOTE: If there is a history of prior malignancy, they must not be receiving other specific treatment for their cancer * History of myocardial infarction =\< 6 months, or congestive heart failure requiring use of ongoing maintenance therapy for life-threatening ventricular arrhythmias * Known primary central nervous system (CNS) malignancy or symptomatic CNS metastases are excluded, with the following exceptions: * Patients with asymptomatic untreated CNS disease may be enrolled, provided all of the following criteria are met: * Evaluable or measurable disease outside the CNS * No metastases to brain stem, midbrain, pons, medulla, cerebellum, or within 10 mm of the optic apparatus (optic nerves and chiasm) * No history of intracranial hemorrhage or spinal cord hemorrhage * No ongoing requirement for dexamethasone for CNS disease; patients on a stable dose of anticonvulsants are permitted * No neurosurgical resection or brain biopsy =\< 28 days prior to registration * Patients with asymptomatic treated CNS metastases may be enrolled, provided all the criteria listed above are met as well as the following: * Radiographic demonstration of improvement upon the completion of CNS-directed therapy and no evidence of interim progression between the completion of CNS-directed therapy and the screening radiographic study * No stereotactic radiation or whole-brain radiation =\< 28 days prior to registration * Screening CNS radiographic study \>= 4 weeks from completion of radiotherapy and \>= 2 weeks from discontinuation of corticosteroids * History of severe allergic, anaphylactic, or other hypersensitivity reactions to chimeric or humanized antibodies or fusion proteins * History or current evidence of bleeding disorder, including bleeding diathesis, i.e., any hemorrhage/bleeding event of Common Terminology Criteria for Adverse Events (CTCAE) grade \>= 2 in =\< 28 days prior to registration * Taking oral prednisone of \>= 10 mg daily or equivalent * Known clinically significant liver disease, including active viral, alcoholic, or other hepatitis; cirrhosis; fatty liver; and inherited liver disease. Notes: * Patients with past or resolved hepatitis B infection (defined as having a negative hepatitis B surface antigen \[HBsAg\] test and a positive anti-HBc \[antibody to hepatitis B core antigen\] antibody test) are eligible * Patients positive for hepatitis C virus (HCV) antibody are eligible only if polymerase chain reaction (PCR) is negative for HCV ribonucleic acid (RNA) * History or risk of autoimmune disease, including, but not limited to, systemic lupus erythematosus, rheumatoid arthritis, inflammatory bowel disease, vascular thrombosis associated with antiphospholipid syndrome, Wegener's granulomatosis, Sjogren's syndrome, Bell's palsy, Guillain-Barre syndrome, multiple sclerosis, autoimmune thyroid disease, vasculitis, or glomerulonephritis. Notes: * Patients with a history of autoimmune hypothyroidism on a stable dose of thyroid replacement hormone are eligible * Patients with controlled type 1 diabetes mellitus on a stable insulin regimen are eligible * Patients with eczema, psoriasis, lichen simplex chronicus of vitiligo with dermatologic manifestations only (e.g., patients with psoriatic arthritis would be excluded) are permitted provided that they meet the following conditions: * Patients with psoriasis must have a baseline ophthalmologic exam to rule out ocular manifestations * Rash must cover less than 10% of body surface area (BSA) * Disease is well controlled at baseline and only requiring low potency topical steroids (e.g., hydrocortisone 2.5%, hydrocortisone butyrate 0.1%, fluocinolone 0.01%, desonide 0.05%, alclometasone dipropionate 0.05%) * No acute exacerbations of underlying condition within the last 12 months (not requiring psoralen plus ultraviolet A radiation \[PUVA\], methotrexate, retinoids, biologic agents, oral calcineurin inhibitors; high potency or oral steroids) * Known active human immunodeficiency virus (HIV) infection (defined as patients who are not on anti-retroviral treatment and have detectable viral load and CD4+ \< 500/ml) * NOTE: HIV-positive patients who are well controlled on anti-retroviral therapy are allowed to enroll * History of idiopathic pulmonary fibrosis, pneumonitis (including drug induced), organizing pneumonia (i.e., bronchiolitis obliterans, cryptogenic organizing pneumonia, etc.), or evidence of active pneumonitis on screening chest computed tomography (CT) scan. Note: History of radiation pneumonitis in the radiation field (fibrosis) is permitted * Severe infections =\< 4 weeks prior to registration, including, but not limited to, hospitalization for complications of infection, bacteremia, or severe pneumonia * History of peripheral neuropathy \>= grade 2 * Known hypersensitivity to docetaxel or polysorbate 80