It's possible to move components between the template and a page.

1. Right-click on the component that you want to move.
2. Select Arrange > Send to template / Send to page.

Tip: You can also copy components over different pages. Select the component you want to copy and type Ctrl/Command + C. Go to the page where you want to move the component to and type Ctrl/Command + V.

Related articles:

How do I add a component to a page in Website Builder?

Which templates can I use in Website Builder?