There could be several issues. Check the following:
With the hotkey N you can make the connections disappear.
The three Net Connection Lines cycle states (through the hotkey N): -Hidden (No connection lines) -Pad to Pad connect -Break (Route to Route)
While dragging the component, cycle through the connections by pressing N. You can view the current mode in the HUD:
If there are a large number of connections, check:
Preferences (gear icon in the top right) ► System ► General ► Advanced button ► Search for PCB.ComponentDrag.ConnectionLimit ► Increase the value to 500.
To have the connection lines move with the component when using Edit ► Move ► Component, as they do when just clicking and dragging the component in the PCB editor, "Comp Drag" must be set to "None". See Preferences ► PCB Editor ► General ► Other (section) ► Comp Drag set to "None". Either way, they will show on the component when you are done.
If tracks have been placed, then for the: None setting: Connection lines will be shown while dragging but the tracks will not move. Connected Tracks setting: Tracks will move with the component.