X[x_,y_, deltat_] := Module[{t,b},
t = x + deltat;
b = y - (1/2)x^2;
{ t , (1/2)t^2 + b}
]

FlowBox[t_] := Module[{Left, Right, Top, Bottom, C, V},
Left = ParametricPlot[ X[Cos[s], Sin[s], t-1], {s, -Pi/2, Pi/2}];
Right = ParametricPlot[ X[Cos[s], Sin[s], t], {s, -Pi/2, Pi/2}];
Top =  ParametricPlot[ X[0, 1, s], {s, t - 1, t}];
Bottom =  ParametricPlot[ X[0, -1, s], {s, t - 1, t}];
C = ParametricPlot[{Cos[s], Sin[s]}, {s, -Pi/2, Pi/2}, PlotStyle->{Red, Thick}];
V = VectorPlot[{1, x}, {x, -2, 2}, {y, -2, 2}, VectorScale-> Small];
Show[{Left, Right, Top, Bottom, C, V}, PlotRange-> {{-2,2}, {-2, 2}}]
]

Animate[ FlowBox[t], {t, 0, 1}, AnimationRunning -> False]