When the relative velocity between the object and the fluid is non-zero, the object will need to displace water in the direction it is moving, because once again it can't occupy the same space as the water at the same time
Displacing the water requires pressure, which, depending on the object's geometry,will generate forces on the water and equal but opposite forces on the object.
Under steady state conditions, i.e. constant velocity (and keeping all other variables fixed, ideally), the forces will be in balance.
The geometry of the object will dictate the direction and magnitude of the forces. Let's ignore drag forces for the time being and only look at lift.
Thought exercise: Using a cube as a candidate object, let's say this cube has neutral buoyancy (i.e. same density as water).
Put this cube under water, and drag it with a zero-width non-stretchy ideal cable attached to the center of the front face at a constant velocity.
As constant velocity, water will flow around the cube. A pressure gradient will exist with the highest pressure in the center of the front face and lowest pressure at the center of the rear face.
Under these conditions the cube will experience no net lift orthogonal to the direction of pull. The lift forces on the left and right will be in balance, as will the lift forces on top and bottom.
(In reality there will be a von karman vortex street behind the cube causing it to oscillate, but let's ignore that for the time being.)
The point of this thought exercise is that, at constant velocity, there will be a pressure gradient determined by the object geometry. The nature of this geometry will determine in which direction the forces act on the object.