I am replacing the upper and lower timing chain on my 98 Altima.

I match the three painted links on the upper chain to the timing marks on each of the three sprockets ( intake cam, exhaust cam and idler ).

I then turn the engine to TDC by hand three or four times ( 2 revolutions for each pass to TDC ). I notice that the painted links no longer match up to the timing marks on the sprockets.

The painted links shift with respect to the timing marks on the sprockets on each turn to TDC. However all three painted links shift by the same amount (counting link pins)

Shouldn't the painted links stay at the same position as I set them originally and match up to the timing marks on each of the three sprockets?

Idk if you fixed this yet but....timing mark on the chain are only for initial instal. only you start cranking the motor the marks will never match again. Just make sure you are at TDC and the camshaft marks line up at first and you should be ok.

If you were at tdc when you took that picture, your timing should be fine. PAAG86 is right- its not worth the work to turn the engine over 25 or more times to get the marks back in alignment.