Key Inclusion Criteria: Study-Wide Eligibility (Across All Study Parts):  Have histologically or cytologically confirmed locally advanced or metastatic NSCLC (Stage IIIB/IIIC or IV) [JACC edition 8], and inclusion of Stage IIIB only if not a candidate for curative therapy.  Must have sufficient tumor tissue (either archived sample or recent biopsy) available for analysis:  Phase 1a Dose-escalation part: EGFR mutations (including but not limited to L858R, exon 19 deletion, T790M, ex20ins, 21exon, G719 X, S7681\L861Q, etc) should be confirmed by previously documented evidence or central lab Phase 1b Dose-expansion part and Phase 2: Have an EGFR in-frame exon 20 insertion test by any central lab Must have at least one measurable lesion as defined by response evaluation criteria in solid tumors (RECIST v1.1)  Prior anti-cancer therapies:  Previously treated with one or more regimens of systemic therapy for locally advanced or metastatic disease Disease progressed, or intolerant to at least one line of systematic therapies including but not limited to any EGFR-target therapies or immunotherapies, for metastatic / local relapsed settings Male or female adult participants (aged 18 years or older, or as defined per local regulations) Eastern Cooperative Oncology Group (ECOG) performance status 0 to 1 (Appendix 2) Estimated minimum life expectancy of 3 months or more  Adequate organ function at baseline  Bone marrow function  Absolute neutrophil count (ANC)≥1.5 x 10^9/L Platelets ≥100 x 10^9/L Hemoglobin ≥9 g/dL, criteria must be met without a transfusion within 2 weeks of the blood draw  Hepatic function  AST and ALT ≤3 x upper limit of normal (ULN); if liver metastases, then ≤ 5 x ULN Bilirubin ≤1.5 x ULN or ≤3 x ULN in the presence of documented Gilbert's Syndrome  Renal function  Creatinine clearance ≥50 ml/min (calculated by Cockcroft and Gault equation (Cockcroft DW, 1976) (Appendix 3)  QTc-related criteria:  A marked baseline prolongation of QT/QTc interval (e.g., repeated demonstration of a QTc interval >480 milliseconds (ms) (CTCAE grade 1) using Fredericia's QT correction formula A history of additional risk factors for Torsades de Pointes (TdP) (e.g., heart failure, hypokalemia, family history of Long QT Syndrome) Willingness and ability to comply with scheduled visits and study procedures  Exclusion Criteria:  Subjects who meet any of the following criteria are not to be enrolled in this study:  Received small-molecule anticancer therapy including cytotoxic chemotherapy, biological agents and investigational agents, ≤21 days prior to first dose of FWD1509 MsOH, or received prior EGFR TKIs (e.g., gefitinib, erlotinib, Osimertinib) ≤7 days prior to the first dose FWD1509 MsOH Have been diagnosed with another primary malignancy other than NSCLC except for patients with adequately treated non-melanoma skin cancer, cervical cancer in situ or definitively treated non-metastatic prostate cancer, or participants with another primary malignancy who are definitively relapse-free with at least 3 years elapsed since the diagnosis of the other primary malignancy Received radiotherapy ≤14 days prior to the first dose of FWD1509 MsOH or have not recovered from radiotherapy-related toxicities. Palliative radiation administered outside the chest and brain, stereotactic radiosurgery (SRS), and stereotactic body radiotherapy are allowed up to 7 days prior to the first dose of FWD1509 MsOH Received strong CYP3A inhibitors and inducers within 2 weeks prior to the first dose of FWD1509 MsOH. They should be discontinued at least 2 weeks prior to the first dose of FWD1509 MsOH and avoided throughout the study duration Received concomitant medications (e.g., statins) which are substrates of BCRP, p-glycoprotein or OATP1B1/1B3 (dose escalation part of phase 1 study) Have undergone major surgery within 28 days prior to first dose of FWD1509 MsOH. Minor surgical procedures, such as catheter placement or minimally invasive biopsy, are allowed  Brain Metastasis: Have known active brain metastases (have either previously untreated intracranial CNS metastases or previously treated intracranial CNS metastases with radiologically documented new or progressing CNS lesions), except for the following conditions:  Brain metastases are allowed if they have been treated with surgery and/or radiation and have been stable without requiring corticosteroids to control symptoms within 7 days before the first dose of FWD1509 MsOH and have no evidence of new or enlarging brain metastases Requiring corticosteroids to control symptoms within 7 days prior to the first dose of FWD1509 MsOH or during study period; patients previously treated for CNS metastases who are clinically stable, have no new lesions, and who do not need treatment with a corticosteroid within the 7 days before the first dose of FWD1509 MsOH and during study period are allowed to be enrolled Have current spinal cord compression (symptomatic or asymptomatic and detected by radiographic imaging) or leptomeningeal disease (symptomatic or asymptomatic) Have significant, uncontrolled, or active cardiovascular disease Have significant, uncontrolled, or active renal disease Have a known history of uncontrolled hypertension (per institution practice); participants with hypertension should be under treatment on study entry to control blood pressure Have any abnormal changes in the cornea or retina that may increase the risk of ocular toxicity during screening Have an ongoing or active infection, including but not limited to, the requirement for intravenous (IV) antibiotics, or a known history of human immunodeficiency virus (HIV), hepatitis B virus (HBV), or hepatitis C virus (HCV); testing is not required in the absence of history Currently have or have a history of interstitial lung disease, radiation pneumonitis that required steroid treatment, or drug-related pneumonitis Female participants who are lactating and breastfeeding or have a positive urine or serum pregnancy test during the screening period Have gastrointestinal illness or disorder that could affect oral absorption of FWD1509 MsOH Have any condition or illness that, in the opinion of the investigator, might compromise participant safety or interfere with the evaluation of the safety of the drug