Does the axis of spin of a curving ball turn as the ball curves? Given a coordinate system, where the $x$- and $y$-axes are on the horizontal plane and the $z$-axis is vertical.
Assume I launch a ball along the $y$-axis with backspin and sidespin. Immediately after being launched, the backspin is around the $x$-axis and sidespin is around the $z$-axis.  As the ball flies through the air, the ball will curve right (or left) due to the sidespin. After flying and curving for a while, the ball will no longer be heading straight down along the $y$-axis as it was when launched. Will the axis of rotation change, for example, to match the one of the ball, or will it still be spinning around the $x$-axis?
 A: Based on the comments, I interpret your question to be whether the axis of rotation changes at all as the ball curves rather than just some component of the axis of rotation changes, which in the original writing I found hard to understand.
The answer to that depends on a couple of things:

*

*What type of ball?

*How precise do you want to be about it?

For most rotating (mostly-spherical) balls in flight that curve, the primary force causing the curved trajectory is called the Magnus force.  It's a result of an interaction between the surface of the ball and the air through which the ball is passing.  For the ball, it will typically be through the center of the ball with a magnitude $F \propto \| \omega \times v \|$, where $\omega$ is the angular velocity vector and $v$ is the linear velocity of the (center of) the ball.  Because this acts through the center of the ball, it will not generate a torque on the ball, and without a torque angular momentum is conserved, which means the axis of rotation does not change.
But if you need to be more precise, the fact is that real balls aren't exactly spherical - in fact most balls for sports deliberately introduce or accentuate some surface features to ensure that they are not spherical.  For example, seams on a baseball are raised, golf balls have dimples, soccer balls are made from panels and so are not quite spherical and have seams.  These deviations may (a) introduce perturbations to the form of the Magnus force or (b) create asymmetric drags on the ball as it rotates and flies.  These differences need not effect different parts of the ball symmetrically because the ball itself is not symmetric, and so they might introduce a torque.  At minimum we would expect a torque that slows the rotation due to friction, but the torque might have components that change the axis of rotation.
I infer that you meant a (mostly) spherical ball, but if you think of oblong balls (e.g. football or rubgy), the axis of rotation can change quite a bit in flight, and thinking of a more extreme case might help with the spherical case that I think interested you.  Footballs may or may not have the nose drop noticeably during flight, a phenomena that depends on several factors including the speed of the ball and the rotational rate of the ball along the long axis at release.  Footballs also sometimes precess in flight, where their axis of rotation is visibly changing during flight but in a (mostly) periodic fashion.  Notably for your question, the precessing ball would happen because of a small amount of what you're calling backspin caused by the ball being thrown with its long axis misaligned to its direction of flight.  So it's not just the spin on the ball but the orientation of the ball at release compared to its direction of flight.
