Transfer apps, data, and settings from Marshmallow to Lollipop

I've been using a Moto E 2nd generation that I bought in Europe, but I've now returned to the US and am replacing it with the US version (XT1527) of the same phone because the European one (XT1524) doesn't get LTE service with my carrier, AT&T.

The old phone is running Android 6.0, but the new one only has Android 5.1, and offered me no options whatsoever to transfer anything over when I set it up.

I'm no Android expert, so forgive me if the answer should be obvious, but…what's the best way for me to get my apps, app data, and settings transferred across to the new phone?

I sort of thought Android was supposed to offer to do this for me at setup, but maybe it doesn't work when moving to an older version of the OS?

Thanks so much!