Note: This is regarding the partner of the receiver, not the receiver themself.
It is possible under the rules to be called a hindrance according to the ITF Rule book
If a player is hindered in playing the point by a deliberate act of the opponent(s), the player shall win the point.
However, the point shall be replayed if a player is hindered in playing the point by either an unintentional act of the opponent(s), or something outside the player’s own control (not including a permanent fixture).
Case 5: In doubles, where are the server’s partner and receiver’s partner allowed to stand?
Decision: The server’s partner and the receiver’s partner may take any position on
their own side of the net, inside or outside the court. However, if a player is creating a hindrance to the opponent(s), the hindrance rule should be used.
As to whether the movement of the partner of the receiver is hindering could be up for debate, but consider this... how far does it go? Could they start jumping up and down waving their arms or yelling out at the time of the serve be considered a hindrance?
In a recreational match, such rules are hard to enforce as there typically is no umpire and it is down to the players themselves being honest and fair.