I took site template with Include content, In that template, there have several custom web parts and CSWP web parts. By using that template I created one site, All the custom web parts, and CSWP added correctly, But In Content Search Web part my Custom Display Templates are not Updated. Default Templates only applied. I don't know why web part properties are not updated, Please Suggest me an Idea about this?
We were running into the same problem, and while I do not have a fix for the display templates in the CSWP we switched everything over to work with a Search Results Web Part which is working for us, just note that there are additional settings for Search Results that that will need to be unchecked to behave like the Content Search.