Actually you are talking about shifting the spin axis rather than a tumble. A tumble will induce a secondary rotation on a second axis.
Very few asteroids are spherical. If they were then there would be insufficient detail to generate a lightcurve. Asteroids, in general, are potatoe shaped and cratered. As such, there is enough 'detail' to detect variations in amplitude as low as 0.02mag, even when viewing the asteroids spin axis directly.
Now a potatoe shaped object will usually result in a bimodal curve. If its a tumbler there will be 2 overlapping periods that results in a 3rd combined period, but it depends on how strong the second axis rotation is. I have done a dual period search on my target and uncovered a possible combination of 287 hr and 130 hr. After subtracting the 130 h period, the 287 h period fits a nice curve, but the 130 h component is a bit ugly.
The primary spin axis is usually through the shortest axis. The secondary component will damp out over time, generally not that much time is required.
I did a bit more research and found another mechanism to slow an asteroids spin rate. Its called angular momentum drain and is the result of a spinning object losing surface ejecta over long periods of time. For my target, the expected dampening limit is between 600 and 700 hrs - gets me halfway
The other interesting thing is that there is no current working model describing what effect collisions may have on spin rates.
Cheers