Details
When copying an entire site, ShareGate Migrate does not automatically copy master pages and page layouts. The reason for this is that master pages and page layouts will not work as-is if they are migrated from one version of SharePoint to another. Unfortunately, ShareGate Migrate does not support updating them so that they work in the new version of SharePoint.
If you are migrating to the same version as the source, the master pages will not automatically maintain their working order. There is a simple workaround for this.
Workaround
Master pages and page layouts can still be migrated to your destination using the Copy Content action:
Connect to the root site of the source site collection and select the Master Page Gallery, then click Next.
Connect to the root site of the destination site collection and select the Master Page Gallery, and click Next.
Select the specific master pages and page layouts to copy, and drag & drop them into the destination.
Note: It is important to only select the elements that you specifically want to migrate, since overwriting all the default master pages and page layouts can easily lead to errors. Also, master pages and page layouts copied between different versions of SharePoint will most likely need to be modified manually in order to work correctly.
If you need to preserve the working order of the master pages while migrating to the same version as the source, copy them the following way:
Create the new, empty, Site Collection at the destination of your migration.
Move the Master pages and page layouts to that site's master page gallery using the Copy Content action.
Use the Copy Structure action to copy the source Site Collection to the one you created in the destination.
This should apply the master pages and layouts properly during the migration.
Note: This will only work if you're copying to the same version of SharePoint. Due to differing code/elements across versions different versions will not behave the same way.