Indeed a beautiful GIF! Inspired by some of Vitaly's ideas, I came up with the following. I used Overlay and created a mask by means of a triangular AlphaChannel "hole" in the image. Hope this helps you to further improve this:

Manipulate[
 Module[{mask, cols},
  cols = RGBColor /@ {"#118DF0", "#ECECDA", "#FF304F", "#0E2F56"};
  mask = Image[
    Graphics[SSSTriangle[1, 1, 1], 
     PlotRange -> {{-.1, 1.1}, {-0.2, 1}}]];
  Overlay[{
    Graphics[{Thickness[.02], CapForm["Round"], 
      Table[{cols[[i]], 
        Circle[SSSTriangle[1, 1, 1][[-1, i]], 
         r, {2 \[Pi] /3 (i - 1), \[Pi] (2 i - 1)/3}]}, {r, s, 1.02, 
        1/n}, {i, 3}]}],
    SetAlphaChannel[Image[Graphics[{}, Background -> cols[[-1]]]], 
     mask]}]],
 {s, 0., 1/n}, {{n, 25}, None}]

This is beautiful, @Clayton Shonkwiler, thank you for sharing! In Graphics3D we have ClipPlanes to cut away portions of a 3D scene from the resulting view. You are right, in 2D (and 3D) we have RegionFunction but then you need to use Plot / Plot3D etc. - plotting functions. On the other hand I think Show and Overlay can combine objects with transparency holes. So you can create a template in a Plot-ing function (via RegionFunction), leading design in Graphics, and then block parts of the design with the template using Show or Overlay. Not sure how practical this is, just dwelling on ideas. Perhaps this might be of use too:

Filling a polygon with a pattern of insets

By lack of better, I am using the following combination of Overlay and SetAplhaChannel to mask images with square or round masks (illustrated with a Manipulate):

Manipulate[Module[{im, sz, mask},
  im = ImageResize[ExampleData[{"TestImage", "Lena"}], 350];
  sz = ImageMeasurements[im, "Dimensions"];
  mask = ImageResize[
    ColorNegate@
     Blur[Image[Graphics[RegularPolygon[pos, r, n], PlotRange -> 1]], 
      blur], Most@sz];
  Overlay[{Switch[bg, "black", mask, "white", Nothing], 
    SetAlphaChannel[im, mask]}]],
 Grid[{
   {Control[{{r, .95, "size"}, .5, 1.25, ImageSize -> Small}], 
    Control[{{blur, 0}, 0, 50, ImageSize -> Small}]},
   {Control[{{n, 4, "shape"}, {3 -> "triangle", 4 -> "square", 
       36 -> "round"}}],
    Control[{{pos, {0, 0}, "position"}, {-.5, -.5}, {.5, .5}, 
      Slider2D, ImageSize -> 70}]},
   {Control[{{bg, "black", "color"}, {"black", "white"}}], 
    SpanFromAbove}}, Alignment -> {Left}, ItemSize -> 22]]

here are some results: