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. Be aware this works only when both webs are on the same domain. If you want to move files between two webs on different domains, try using a SharePoint 2013 workflow and the **call web service** action.

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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).