Migration testing equals water (or other solvent) extractive testing; there is no difference at the general level.
The EU regulations use the word 'migration', the US FDA rules don't use the word migration.
Migration/extractives testing works by putting the food contact item (eg. packaging) into a bath of water or other solvent for a certain amount of time (and temperature) to see what chemicals migrate out of the item into the solvent. The solvent is then evaporated away and the amount of chemical that migrated out of the item is measured. If a lot of chemicals come out of the item and end up in the solvent then obviously the same might happen when that item is in contact with food. Not good. Hence the limits for 'extractives' for food contact materials like packaging.
Under US FDA rules anything that contacts food has to meet composition requirements and also pass one or more extraction tests (=migration tests). The solvent to be used for the extraction depends on the intended use of the food contact article. The time and temperature of the extraction also has to be chosen correctly. So the testing requirements will be different depending on whether the packaging will be used for dry solid foods or for acidic liquids. There are also different requirements for time and temperature depending on whether (for example) the packaging will be used for hot fill processes or retorting.
The relevant rules are: US FDA Code of Federal Regulations 21, Chapters 174 - 177.
Chapter 177 is for polymers, while the other chapters are for paper packaging and coatings. Here's a link to chapter 177:
In my experience a water-only extraction probably isn't enough to be sure that the packaging is safe. It might not meet the FDA rules either. You will need to check the rules as they relate to the intended use of the packaging.