What is the relationship between Ekman transport and Ekman pumping?
The link is the conservation of mass equation: $$ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0. $$
Briefly, Ekman pumping is the horizontal divergence of the Ekman transport. 

If we integrate this expression over the Ekman layer,
$$ \int_{-D}^0 \frac{\partial u}{\partial x} dz + \int_{-D}^0 \frac{\partial v}{\partial y} dz + \int_{-D}^0 \frac{\partial w}{\partial z} dz = 0$$ 
where $D$ is the height of the Ekman layer we take $z=0$ to be the surface.  This gives
$$ \frac{\partial}{\partial x} \int_{-D}^0  u dz +  \frac{\partial }{\partial y} \int_{-D}^0 v dz + w_{z=0} - w_{z=-D} = 0$$ or,
$$ \frac{\partial}{\partial x} U +  \frac{\partial }{\partial y} V + w_{z=0} - w_{z=-D} = 0$$ 
where $U$ and $V$ are the horizontal volume (Ekman) transports.  As a boundary condition we take $w_{z=0} = 0$ leaving
$$ w_{z=-D} = \frac{\partial}{\partial x} U +  \frac{\partial }{\partial y} V. $$
So, the Ekman pumping velocity is the horizontal divergence of the Ekman transport (and thus also proportional to the curl of the wind stress).