Tread lightly laddie, there be beasties in these woods.
In a 6-axis arm, the payload (mass, not weight or force) is directly related to the torque carrying capacity of the Joint5 wrist and Joint6 ToolFlange axes. These are the two weakest joints in the entire arm structure. Therefore the robot manufacturer provides mass moment of inertia limits for J5 & J6 for dynamic motion. Dynamic motion is analyzed with the equation Torque = MassMomentOfInertia X AngularAcceleration, or T = J X Alpha. The manufacturer's published value of payload is valid for maximum payload & maximum speed & all arm configurations. Again, payload is "kg", which is mass not force or weight.
Usually manufacturers will also publish maximum torque values for these axes.
The total torque applied to J5 & J6 is determined by the dynamic loading + torques due to any static loading (gravity effects) + torques due to any externally-applied forces. You must perform a free body diagram analysis of your motion configurations and determine your worst case applied torque to the J5 & J6.
If you want to use the robot to "push" an object, you must analyze the arm configuration to determine worst case applied torques. Primary concern would be J5 & J6, but you may also need to analyze for J4, J3, & J2. This would apply even for an EOAT with a pneumatic or spring pusher action since that reactive force would be transmitted back to the arm links.
That is, unless you develop a sophisticated EOAT that mechanically locks to a fixture so that the locking mechanism handles the induced forces. Then the robot is used for what it is designed to do: move objects from point to point, not pushing.
On a heavy-payload arm, your applied push-force resultant torques and loads probably wouldn't matter. On a smaller-payload arm, they may be significant.
After all of that, you might be limited by the controller and it's motor protection algorithms. The pushing EOAT must be located at a TargetPosition. In order to apply a "pushing" force, the arm must be commanded to move to a new position TargetPosition + Delta. But if the robot controller commands the robot to move the EOAT to TargetPosition+Delta and the encoders report that it does not arrive at that TargetPosition+Delta position within a specific amount of time, then it will likely issue a servo fault.