I am using currently InDesign CC ME version. While creating new document and applying custom Paragraph Styles (with Adobe Paragraph Composer and LTR direction) everything looks ok, until I will open/create new document and paste that text there, I have to clear local overrides to bring back the LTR direction of the text.
Setting global default composer in InDesign preferences doesn't seem to work either.
Check the composer settings for each document. If the document was created before you set the global preference, changing that won't effect it. They probably each have different composers.
What paragraph style Is applied to the text? If there is an identically named style in the target document, the pasted text will inherit that style definition. So, if you have the text marked as "Basic Paragraph" in the source, and it is RTL in the style in the source doc, but the target doc has the Basic Para style set up as LTR, then the pasted text will obey the LTR specified in the target document's styles.