Inclusion criteria:  Signed and dated, written informed consent form (ICF) prior to any trial-specific procedures Male or female aged ≥ 18 years (no upper limit of age) at the time of ICF signature Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 Life expectancy of at least 3 months Initially, only patients with a Signal Regulatory Protein-alpha (SIRPα) polymorphism V1/V1 will be eligible; SIRPα polymorphism will be assessed in blood sampling (patient deoxyribonucleic acid (DNA)) in a central laboratory; V1 allele is understood to include V1 and potential V1-like alleles. If, at a later time, V1/V2 heterozygous patients are considered for inclusion in this trial, these patients will require to be centrally confirmed with at least one V1 allele. Patients with histologically or cytologically documented advanced/metastatic primary or recurrent Head and Neck Squamous Cell Carcinoma (HNSCC), melanoma, Non-Small Cell Lung Cancer (NSCLC) who failed or are not eligible to standard therapy Patients with at least one measurable lesion are allowed as per Response Evaluation Criteria In Solid Tumors (RECIST) v1.1 Patient must have at least one Positron Emission Tomography (PET) imageable and evaluable tumor lesion with a diameter of at least 20 millimeter Further inclusion criteria apply.  Exclusion criteria:  Patients with symptomatic/active central nervous system (CNS) metastases; patients with previously treated brain metastases are eligible if there is no evidence of progression for at least 28 days before the first study treatment administration, as ascertained by clinical examination and brain imaging (Magnetic Resonance Imaging (MRI) or Computed Tomography (CT)) during the screening period Other tumor location necessitating an urgent therapeutic intervention (e.g., palliative care, surgery or radiation therapy, such as spinal cord compression, other compressive mass, uncontrolled painful lesion, bone fracture) Presence of other active invasive cancers other than the one treated in this trial within 5 years prior to screening (or less, pending discussion with sponsor), except appropriately treated basal cell carcinoma of the skin, or in situ carcinoma of uterine cervix, or other local tumors considered cured by local treatment Patients with active autoimmune disease or a documented history of autoimmune disease, that requires systemic treatment (i.e. corticosteroids or immunosuppressive drugs); except patients with vitiligo, resolved childhood asthma/atopy, alopecia, or any chronic skin condition that does not require systemic therapy, patients with autoimmune-related hypothyroidism on a stable dose of thyroid replacement hormone and/or controlled Type 1 diabetes mellitus on a stable insulin regimen may be eligible Known severe infusion related reactions to monoclonal antibodies (Grade ≥ 3 National Cancer Institute (NCI)- Common Terminology Criteria for Adverse Events (CTCAE) v5.0) and patients removed from previous anti-Programmed-cell-death-protein-1 (PD-1) or anti-Programmed-cell-death ligand-1 (PD-L1) therapy because of a severe or life-threatening immune-related adverse event (irAE) (Grade ≥ 3 NCI-CTCAE v5.0) Patients receiving systemic treatment with any immunosuppressive medication within one-week prior to treatment start with BI 765063 and ezabenlimab; steroids of max. 10 mg prednisolone equivalent per day are allowed, topical and inhaled steroids are not considered as immunosuppressive Patients who have interstitial lung disease or active, non-infectious pneumonitis. Patients with uncontrolled disease-related metabolic disorders (e.g., hypercalcemia, Syndrome of Inappropriate of AntiDiuretic Hormone Secretion (SIADH)) or uncontrolled diabetes Further exclusion criteria apply.