I ended up using a Visual Basic for Applications solution with this code, which should be adaptable to other people's uses if anyone reads this qustion later:
Sub OnSlideShowPageChange()
If ActivePresentation.Slides(ActivePresentation.SlideShowWindow.View.CurrentShowPosition).Name = "Countdown" Then
Dim obj As ShockwaveFlash
Set obj = ActivePresentation.Slides("Countdown").Shapes("Timer1").OLEFormat.Object
obj.Playing = True
obj.Rewind
obj.Play
End If
If ActivePresentation.Slides(ActivePresentation.SlideShowWindow.View.CurrentShowPosition).Name = "Countdown2" Then
Dim obj2 As ShockwaveFlash
Set obj2 = Nothing
Set obj2 = ActivePresentation.Slides("Countdown2").Shapes("Timer2").OLEFormat.Object
obj2.Left = -999
MsgBox ("Its x position is now " & obj2.Left)
End If
End Sub
I named two slides which had Flash movies on them as "Countdown" and "Countdown2" and the two movies as "Timer1" and "Timer2". The code below resets the critical properties of those movies when the specified slides are entered.