Firstly, there are no SharePoint REST APIs working specifically with the **SP.RequestExecutor** object. This Javascript object is used inside SharePoint-hosted apps to make web service calls (using REST APIs or JSOM) outside the SharePoint apps.
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Secondly, from you question, I am not sure if you want to move files from:

 - a SharePoint-hosted app's web to the host web => If you want to move files from the app web to the host web using JS, there is no other way than using the **SP.RequestExecutor** object.

 - or from a standard SharePoint web to another standard web => In this case you can simply use AJAX (jQuery, SuperAgent, etc.) and the RESTful URL below.

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Here is the Uri structure for the **SP.File.moveTo** method: `http://<sitecollection>/<site>/_api/web/folders/getbyurl(folderrelativeurl)/files/getbyurl(filerelativeurl)/moveTo(newUrl,flags)` as taken from [here](https://msdn.microsoft.com/EN-US/library/office/jj245487.aspx).