The Marangoni Effect says that fluid will want to flow from areas of lower surface tension to areas of higher surface tension.
Soap has a lower surface tension than Water/ Milk. And as a result, when soap is placed on the surface of a fluid (as it is, in these animations), it wants to flow away to areas of higher surface tension. A more in-depth explanation of the Marangoni effect and surface tension in general can be found a previous ZME Science post.
And this propels the small boat, causes the pepper flakes to spread away, makes the string to expand, and the dye to fan out. It is also responsible for the Tears of Wine phenomenon that you might have already witnessed. : )
PC: Flow Visualization at UC Boulder, source video, MIT, Dan Quinn