If a batsman moves in his crease, right or left, does the wide also moves with him? If a right-handed batsman moves to his right and leave the stumps and the bowler bowls a ball which is in the reach of the batsman but outside the wide line, Will this be a wide call or not?
Basically, yes. If a batsman moves, the ball will based on both their current position and their normal position. Quoting from Law 22 - Wide ball:
22.1.1 If the bowler bowls a ball, not being a No ball, the umpire shall adjudge it a Wide if [...] the ball passes wide of where the striker is standing and which also would have passed wide of the striker standing in a normal guard position.
The effect of this is that a batsmen moving expands the area into which a bowler can bowl rather than "moving" the area.