1978 KZ650, I have no idea what I'm doing.

Bearings appear readily available. Found these which includes the sleeve, which is actually the center race.

http://www.ebay.co.uk/itm/KAWASAKI-KZ-650-D1-D2-SR650-1978-1979-SWING-ARM-NEEDLE-BEARING-SET-/271327518082

The center race or sleeve actually serves another enormously important function. It is an important structural component of your chassis, and an understanding of what it does and how it works is important to being able to assemble it properly.

Consider its installation without the swing arm and bearings for a moment. The assembly consists of the two end caps and center sleeve between the two sides of the frame. Those components are clamped tightly together making a very strong and rigid element of the chassis. The center sleeve does not spin, and becomes very much like another tube welded into the frame structure. The swing arm simply spins on the center sleeve, and is trapped in between the two end caps. Radial play is minimized by the close tolerances between the bearing shells and rollers, but those bearings do nothing to control axial play. Imagine if the swingarm were narrower than the space between the end caps; the swingarm would slide back and forth along the center sleeve. If the width of the swingarm is precisely the same as the center sleeve, then there will be zero axial play, which is what you want. Production tolerances being what they are, the sleeve has to be a bit longer than the swing arm width to ensure it does not end up being shorter which would cause the swing arm to be rigidly clamped inside the assembly. The factory makes up for this tiny extra length by adding shims under the end caps. Usually these are very thin, and easily overlooked. You may have some embedded in the old grease inside your end caps. It is very rare in my experience to be able to buy shims even at the dealer. They are even often left off of the parts list! But they are usually needed to achieve a good assembly. You should be able to get them at a good bearing supply. I usually have some on hand as I always make sure to scavenge them when I have an opportunity.

Proper assembly is a trial and error process. I dry assemble everything and torque the pivot bolt. Then check for play. Disassemble and add shims (I guess at how much I think is needed) and re-assemble and torque the bolt. You should end up with absolutely no perceivable slop. If the swing arm is getting pinched, take out shims or replace with thinner. Once it is right, take it back apart and grease everything up and re-assemble. You should have a fairly stiff assembly - the bare swingarm should be held tightly enough that it barely falls by its own weight.
 
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