Symptoms
You are migrating a site from a source to a destination, but your target audience web part does not appear in the destination.
Details
Target audience web parts are not supported for migration.
Solution
If a list or library uses targeted audiences, the values can be preserved during migration if the audiences are created manually before the migration takes place, and activated manually at the destination. For more on creating audience, please see here.
Please note that not all target Web Part audiences are supported, so it is best to verify in the destination.