What is the ruling on initiating or returning greetings to non-Muslims, such as "Good morning," with the intention of winning over their hearts to invite them to Islam, and does this continue after they have not responded to the invitation?

It is permissible to return a greeting from a non-Muslim in kind, such as saying "Good morning" if it is said to you. It is also permissible to initiate such greetings with them, especially for the purpose of winning their hearts over and inviting them to Islam. There is no harm in gifting them da'wah booklets and other materials with the aim of winning their hearts and encouraging them towards Islam. Continuing to treat them kindly may influence them, even if they do not respond immediately.