The purpose of this trial is to determine which doses of Lu AG09222 are recommended to help prevent migraines. People who join this trial have already tried 1 to 4 other available medications to prevent their migraines, but these medications have not helped them.

Key Inclusion Criteria:

• The participant has a diagnosis of migraine as defined by International Classification of Headache Disorders Third Edition (ICHD-3) guidelines confirmed at the Screening Visit.
• The participant has a history of migraine onset ≥ 12 months prior to the Screening Visit.
• The participant has a migraine onset at ≤50 years of age.
• The participant has ≥4 migraine days per month for each month within the past 3 months prior to the Screening Visit.
• The participant has documented evidence of treatment failure in the past 10 years of at least 1 to 4 (maximum) different migraine preventive medications.

Key Exclusion Criteria:

• The participant has previously been dosed with an anti-pituitary adenylate cyclase-activating peptide (anti-PACAP treatment).
• The participant has confounding and clinically significant pain syndromes.
• The participant has a diagnosis of acute or active temporomandibular disorder.
• The participant has a history or diagnosis of confounding headaches.

Additional protocol-defined criteria apply.