Crosspicking is a specific type of flatpicking.
Flatpicking is merely the technique of playing runs of single notes using a flatpick. (Technically we also strum with a flatpick, but usually flatpicking is seen as an alternative to strumming.)
Crosspicking involves the use of flatpicking patterns across several strings. Check out this demonstration of Molly Tuttle crosspicking "Wildwood Flower." You can see that by imposing her picking pattern across several strings, she adds rhythmic complexity while not only carrying the melody, but also offering harmony (chords) as well, without strumming at all.
If you are just starting out, you will want to master the basic back-and-forth picking of flatpicking before getting into the more complex rhythmic patterns of crosspicking. If you search these terms on the internet, you'll find lots of stuff for guitarists, a little for mandolin players, and almost nothing for dulcimer players. Among dulcimer players, the most prolific at both would be Aaron O'Rourke and Gary Gallier. Stephen Seifert is clearly capable of the technique as well, as is Erin Mae, but they tend to strum a lot more than is usually included in crosspicking.
Edit: In that Molly Tuttle video, she doesn't play the song until 12:52. She begins teaching it at 7:48. The first 7+ minutes is her explaining the crosspicking pattern she uses.