A turntable seen from above, free to spin about a fixed axis, carrying two equal masses — one on each side. Move them from the starting radius to the finishing radius while nothing outside exerts a torque about the axis: the moment of inertia changes, the spin rate changes to compensate, and the angular momentum does not change at all. The two bars in the drawing show how that unchanged total is split between the turntable and the two masses — the split moves, the end of the bar does not. The rotational energy is a different story, and the note under the drawing says where it goes.
WhyMoving the two 3.00 kg masses from 0.80 m to 0.20 m changes the moment of inertia from 5.840 to 2.240 kg m2, so the spin rate goes from 2.50 to 6.518 rad/s while L stays 14.60 kg m2/s.
The split
What to noticeAngular momentum is conserved because nothing outside the system exerts a torque about the axis. The rotational energy is not: it rose by 29.33 J, and that is the work your arms did pulling the masses inward.