Animating moving surface of torus

Let's get a black torus:

torus = First@ParametricPlot3D[{Cos[u] (3 + Cos[t]), Sin[u] (3 + Cos[t]), Sin[t]},
                               {u, 0, 2 Pi}, {t, 0, 2 Pi},
                               PlotStyle -> Black, Mesh -> None, PlotPoints -> 10]

and now, this is a way to go:

DynamicModule[{d1 = 0, d2 = 0},
 Column[{
   Graphics3D[{
     torus,
     Red, Dynamic[Riffle[
       Point /@ Array[
         {Cos[#] (3. + 1.01 Cos[#2]), Sin[#] (3. + 1.01 Cos[#2]), Sin[#2]} &,
         {65, 15},
         {{0 + d2/10, 2. Pi + d2/10}, {0. + d2, 2 Pi + d2}}]
       , {Yellow, Pink, LightBlue}
       ]]
     }
    , ImageSize -> 500, Background -> Black, Boxed -> False]
   ,

   Slider[Dynamic@d2, {0, 2. Pi, .01}]
   }]]

Not perfect but I don't have time now for more efficient approach :/.

p.s. Array works this way on version 9. Use Table/Range for older versions.

Here's my take:

which was produced by

torus[c_, r_] := BSplineSurface[Map[Function[pt, Append[#1 pt, #2]],
                                {{1, 0}, {1, 1}, {-1, 1}, {-1, 0},
                                 {-1, -1}, {1, -1}, {1, 0}}] & @@@
                                (TranslationTransform[{c, 0}] /@
                                 (r {{1, 0}, {1, 1}, {-1, 1}, {-1, 0},
                                     {-1, -1}, {1, -1}, {1, 0}})),
                                SplineClosed -> True, SplineDegree -> 2, 
                                SplineKnots -> ConstantArray[{0, 0, 0, 1/4, 1/2,
                                                              1/2, 3/4, 1, 1, 1}, 2], 
                                SplineWeights -> Outer[Times, {1, 1/2, 1/2, 1,
                                                               1/2, 1/2, 1},
                                                              {1, 1/2, 1/2, 1,
                                                               1/2, 1/2, 1}]]

toreq[c_, r_][u_, v_] := N[{(c + r Cos[v]) Cos[u], (c + r Cos[v]) Sin[u], r Sin[v]}]

With[{c = 3, r = 1, m = 37, n = 17}, 
     Animate[Graphics3D[{{Black, torus[c, 0.98 r]}, 
                         Point[RotationTransform[2 φ, {0, 0, 1}] @
                               Flatten[Table[toreq[c, r][u, v + φ],
                                             {v, 0, 2 π, 2 π/(n - 1)},
                                             {u, 2 π, 0, -2 π/(m - 1)}], 1], 
                               VertexColors ->
                               Flatten[Table[{Cyan, Magenta, Yellow}[[Mod[k - j, 3] + 1]],
                                             {j, n}, {k, m}]]]},
                        Background -> Black, Boxed -> False],
             {φ, 0, 2 π, 2 π/(4 (n - 1))}]]