Let's look at it like a math problem.

Suppose there is X in US bonds, Y in US equities, and Z in European bonds.

Now, suppose there becomes a better yield in US bonds, so the Z money goes into US bonds. U.S. bond prices go up (and thus yields go down) because of demand, of course.

Why does the Y money also go into bonds?

Bonds up, stocks down is a nice simple rule of thumb, buy why does it apply in this case, when a bunch of new money comes into the game?