This is a multicenter, open-label, single-arm study to investigate the safety, tolerability, PK, pharmacodynamics and preliminary activity of INCA32459 in participants with selected advanced malignancies. Part 1 (dose escalation) will determine the recommended dose of INCA 32459 for expansion (RDE) and the maximum tolerated dose (MTD). Part 2 (dose expansion) will further evaluate the safety, tolerability, PK, pharmacodynamics, and preliminary antitumor activity of INCA 32459 at the recommended dose(s) for expansion in 2 tumor-specific cohorts.

Inclusion Criteria:

• Histologically or cytologically confirmed advanced malignancies as follows:

  1. Part 1 only: Participants with the select advanced malignancies as specified in the protocol.

  2. Part 2 only:

     • Cohort 1 only: Participants with Stage III (unresectable) or Stage IV (metastatic) melanoma that is considered nonamenable to curative treatments or procedures.
     • Cohort 2 only: Participants with histologically or cytologically confirmed recurrent/metastatic SCCHN that is PD-L1 positive (CPS ≥ 1) which is not amenable to local therapy with curative intent.

• Participants must have experienced disease progression after treatment with standard therapies, or are intolerant to or ineligible for standard treatment:

  1. Part 1: All available standard therapies, including anti-PD-(L)1 and platinum-based therapy, if applicable, that are known to confer clinical benefit. Prior anti-PD-(L)1 therapy should not have been discontinued because of intolerance.
  2. Part 2: Available standard therapies, including anti-PD-(L)1 and platinum-based therapy, if applicable, that are known to confer clinical benefit. Prior anti-PD-(L)1 therapy should not have been discontinued because of intolerance. Part 2 participants may have received up to 2 prior systemic therapies in the a advanced/metastatic setting.

• ECOG performance status of 0 or 1

• Part 2 only: Measurable disease according to RECIST v1.1.

• Part 2 only: Willingness to undergo a fresh tumor biopsy at screening (core or excisional).

• Part 2 only: Willingness to undergo a fresh tumor biopsy at screening and on-treatment in selected participant.

• Willingness to avoid pregnancy or fathering children

Exclusion Criteria:

• Prior treatment with any LAG-3- or MHC Class II-directed therapy for current malignancy, or any prior malignancy.
• Treatment with anticancer therapies or participation in another interventional clinical study within 28 days before the first administration of study treatment (this includes curative radiation to the thorax or systemic anticancer therapies).
• Not recovered to ≤ Grade 1 or baseline from residual toxicities of prior therapy (with exceptions specified in the protocol).
• Not recovered adequately from toxicities and/or complications from surgical intervention before starting study treatment.
• Palliative radiation therapy administered within 1 week of first dose of study treatment or radiation therapy in the thoracic region that is > 30 Gy within 6 months of the first dose of study treatment.
• Any known additional malignancy that is progressing or requires active treatment; history of other malignancy within 3 years of the first dose of study treatment (with exceptions specified in the protocol).
• Evidence of interstitial lung disease or history of interstitial lung disease, or active, noninfectious pneumonitis.
• Active autoimmune disease requiring systemic immunosuppression with corticosteroids (> 10 mg/day of prednisone or equivalent) or immunosuppressive drugs within 2 years before the first dose of study treatment.
• Untreated brain or CNS metastases or brain or CNS metastases that have progressed (eg, evidence of new or enlarging brain metastasis or new neurological symptoms attributable to brain or CNS metastases).
• Chronic treatment with systemic steroids (> 10 mg/day of prednisone or equivalent).