How can I change the text position of connector?

As of Visio 2016 use the Text Block function

I'd rather use the old method, that is predefining where all the labels be automatically, but it seems the whole text block related to the figure (we call labels) are considered a separate object.

Try this out:

Move or rotate text On a shape On the Home tab, in the Tools group, click the Text Block tool . Click the shape to select its text block. NOTE If the text is in a group, click to select the group, and then click to select the text in the shape.

Drag the text to move the text block. On the Home tab, in the Tools group, click the Pointer Tool . After you switch back to the Pointer Tool , the text keeps the same position relative to the shape. If you use the Pointer Tool to drag the text, the shape will also move, even if the text is no longer on the shape. To move the text independently of the shape, use the Text Block tool .

http://office.microsoft.com/en-us/visio-help/add-edit-move-or-rotate-text-on-shapes-HA102749355.aspx

Work around.

Searched for Dynamic Connector in 'Search for Shapes'. 100s of dynamic connectors popped up. Found the yellow handle for moving text to be present in 'Dynamic connector.44'. Boom.