Is there any setting that could be used to improve the results of TransformedRegion? For some transformations it appears to perform very poorly by default. Below is one example taken from MMA.SE (the transformation function specifically is written by Michael E2).

Transformation function:

flow = Module[{x, y}, Function[{t, a, b},
    Evaluate[
     {
       x[t, a, b], y[t, a, b]} /. First@NDSolve[{
         D[x[t, a, b], t] == y[t, a, b],
         D[y[t, a, b], t] == -Sin[x[t, a, b]],
         x[0, a, b] == a,
         y[0, a, b] == b
         },
        {x, y},
        {t, 0, 20},
        {a, -1/2, 1/2},
        {b, 1/2, 3/2}
        ]
     ], Listable]];

Region:

object = Disk[{0, 1}, 0.5];

Background:

splot = StreamPlot[{y, -Sin[x]}, {x, -4, 4}, {y, -3, 3}, StreamColorFunction -> "Rainbow"]

Somewhat correct transformations given by ParametricPlot for $t=0$ and $t=10$:

Show[
 splot,
 ParametricPlot[
  flow[t, r Cos[theta], 1 + r Sin[theta], t],
  {r, 0, 0.5},
  {theta, 0, 2 Pi}
  ]
 ]

Incomplete transformations given by TransformedRegion:

Show[
 splot,
 RegionPlot[
  TransformedRegion[
   object,
   flow[t, #, #2] &
   ]
  ]
 ]