If you are looking to share an app template with other Fulcrum users, please check out our How do I share an app template I've built? article.
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However, if you are attempting to share an app template among the organizations associated with your account, in order words, transfer it, then your best choice would be to export the app template and then import it into the desired organization. Here is how you do it.

First, go to the app dashboard of the app you want to export. Then add /export at the end of the URL, as shown in the gif below.
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Once you hit the Enter key, a download will automatically start. The name of the generated file will vary according to your App's Name, however, its extension is always going to be .fulcrumapp ({yourAppName}.fulcrumapp). The generated file will be saved in your Downloads folder.

Now that you have your App Template downloaded, switch to the other organization. Make sure you are on the Apps screen, and add /import at the end of the URL.

Choose the app template file you just downloaded and hit the CREATE APP button. You will be redirected to the App Designer of the app you just imported!

And that is how you transfer an App Template among your different organizations.