Update page layout before Thread.Sleep()
<Grid
  x:Name="ContentPanel"
  Grid.Row="1"
  Margin="12,0,12,0">
  <TextBlock
    Margin='79,263,177,307'
    Name='textBlock1'
    Text='TextBlock'
    RenderTransformOrigin="0.5,0.5">
    <TextBlock.RenderTransform>
        <CompositeTransform />
    </TextBlock.RenderTransform>
  </TextBlock>
  <Button
    Content="Button"
    Height="80"
    Margin="116,0,188,144"
    VerticalAlignment="Bottom"
    Click='Button_Click' />
</Grid>

<phone:PhoneApplicationPage.Resources>
<Storyboard
  x:Name="MoveTextBlockStoryboard">
  <DoubleAnimationUsingKeyFrames
    Storyboard.TargetProperty="(UIElement.RenderTransform).(CompositeTransform.TranslateX)"
    Storyboard.TargetName="textBlock1">
    <EasingDoubleKeyFrame
      KeyTime="0"
      Value="0" />
    <EasingDoubleKeyFrame
      KeyTime="0:0:1.1"
      Value="120" />
  </DoubleAnimationUsingKeyFrames>
  <DoubleAnimationUsingKeyFrames
    Storyboard.TargetProperty="(UIElement.RenderTransform).(CompositeTransform.TranslateY)"
    Storyboard.TargetName="textBlock1">
    <EasingDoubleKeyFrame
      KeyTime="0"
      Value="0" />
    <EasingDoubleKeyFrame
      KeyTime="0:0:1.1"
      Value="-105" />
  </DoubleAnimationUsingKeyFrames>
</Storyboard>

private void Button_Click(object sender, RoutedEventArgs e) {
  textBlock1.Text = "new text";
  MoveTextBlockStoryboard.Begin();

}