FMX doesn't auto-capture the control the mouse went down on. But you can do this manually, either by TForm.MouseCapture (that's for the form itself) or with TForm.SetCaptured(AnyControl) - this is for a particular control on the form.
procedure TForm1.FormMouseDown(Sender: TObject; Button: TMouseButton; Shift: TShiftState; X, Y: Single);
begin
if (Sender <> self) then
SetCaptured(TControl(Sender))
else
MouseCapture;
memo1.lines.add(Sender.Classname + ' mouse down');
end;
procedure TForm1.FormMouseUp(Sender: TObject; Button: TMouseButton; Shift: TShiftState; X, Y: Single);
begin
ReleaseCapture;
memo1.lines.add(Sender.Classname + ' mouse up');
end;
Setting the mouse capture is important if you do anything in MouseDown that interacts with the control being clicked. Without capture, the user may click (mouse-down) on Panel1, move it, and release it on Panel2. The MouseUp will come from Panel2 in this case, or no mouse-up at all. By setting the mouse capture, you are guaranteed that you'll receive a MouseUp from that control that is being captured. VCL does this automatically, FMX does not.
If you don't need the MouseDown on form level, but just for particular controls, use TControl.AutoCapture instead.