Licenses Tied to Home Console: On 360, licenses bind to console + profile. On One/Series, "Home Xbox" setting manages sharing. Set new console as Home to share offline
Xbox 360 Marketplace Purchases: The 360 store closed July 2024. Buy BC games on the modern store before then if delisted
Setup
Sign in with same email/password (or Authenticator) on new console
Go to Settings → General → Personalization → My home Xbox → "Make this my home Xbox"
Install games from "Full Library" → "Owned games" → Filter "Xbox 360"