Effective field in MRI when solving for Bloch equations
In the rotating reference frame, $B_z$ from the static magnetic field $B_0$ is transformed away, hence $B_z = 0$ if no gradient field is applied. All gradients change the $B_z$ value, as you described it with the slice selection gradient. This gradient causes $B_z$ to increase when you move along $z$. Note that it is irrelevant if you do this from the center of the magnet or somewhere close to the bore - you will have the same $z$-dependence at each position $(x,y)$.

For the readout-gradient (assuming it is along left-right direction that we'll call $x$), the same is true: $B_z = G_Rx$. If you go from left to right, $B_z$ changes. In this case, it is independent of the position along the bore, so moving along $z$ does not have an influence on the value, if you stay at the same $x$ coordinate.

A gradient field always acts along the $z$-component of the magnetic field. This is always a bit tricky, since gradients can be "applied along each direction".