A constant current means, for an ideal motor, a constant torque. This is approximately true for real motors. It doesn't matter what you attach to the motor, or how fast it's turning.
What you seem to be missing is Newton's second law of motion. It states that force is the product of mass and acceleration:
$$ F = ma $$
The constant current you supply to the motor is one force. The weight opposes that force. The difference is the net force, \$F\$ in this equation, and \$m\$ is the mass of the weight, plus the mass of the rotor and the string and everything else the motor must move.
You set current to be sent to the motor so that the torque applied is 10 in-lbs without any load.
Not possible. There is nothing for the motor to "torque against". This is the mechanical equivalent of trying to develop 10 volts across a dead short. The motor will rapidly spin at its maximum speed, and the back-EMF will rise to the driving voltage such that your driving electronics are unable to supply enough voltage above the back-EMF to make enough current to have that much torque.
Let's just say you determine how much current is required for 10 in-lbs of torque, and you drive your motor with a constant-current supply set to that.
What happens when the torque from the weight/load is 5 in-lbs?
Assuming that the rotor and the string are massless and frictionless, the weight will be accelerated upwards by the net 5 in-lbs of torque (motor's 10 in-lbs, less 5 in-lbs from the weight). The rate of the acceleration is determined by the mass of the weight and Newton's law above.
As the speed of the motor changes (the weight is accelerating), the back-EMF also changes. Your constant-current supply to the motor will have to apply an increasing voltage to maintain the same current. Electrical power thus goes up, as does mechanical power.
What happens when the torque from the weight/load is 10 in-lbs?
Motor torque balances weight torque. However fast the weight is moving (if at all), it keeps doing that. Newton's first law applies.
What happens when the torque from the weight/load is 15 in-lbs?
The weight will accelerate downward, overpowering the motor. However, it won't be a free-fall. The motor cancels some of the force of the weight, resulting in a slower acceleration downwards.
If the weight overpowers the motor, then eventually it can get the motor to run backwards, relative to the way it would run if there were no load. When this happens, the back-EMF now adds (instead of subtracts) from the voltage you apply to the motor. At some point, your controller, which is attempting to maintain a constant current, must apply a negative voltage to maintain that current. In other words, the back-EMF is sufficient to create the necessary torque on its own: your controller must oppose it.
This is perfectly symmetrical with the first case, where the motor was overpowering the weight. In that case, electrical and mechanical power went up (without bound, if you let them). In this case, electrical and mechanical power go down (negative, if you let them). Energy is conserved because you are changing the gravitational potential of the weight.
The need to resist the back-EMF usually means storing electrical energy in a capacitor or battery, or using it to heat a resistor. If you can't do this fast enough, then the motor will create more torque than your desired 10 in-lbs, and you have hit the limits of your "constant current" driver.
Further reading:
Raising the frequency will not help you. Even at 100 Hz, the mechanical inertia of the motor will make it run smoothly. The only difference will probably be slightly more heat being generated, as the power transistor is switching ON/OFF more frequently.
Duty cycle modulation seems quite doable. The only slight problem could be that the load on the motor is not constant - at the max excursion of the membrane, probably the pressure on the motor is higher. So, at low duty cycle, the motor might refuse to turn, while still consuming energy. You can probably avoid that with a minimum duty cycle.
Best Answer
You cannot expect two open-loop motors to have the same performance - even if matched initially they may age differently. I'd especially expect that with brushed DC motors where brush wear can be a part of aging, or any system with sleeve-type bearings.
You can used closed loop control monitoring a rotary encoder to slave both motors to a desired rotation rate, or one to the other other, or switch to stepper motors, however electronics may not be the best solution.
Instead, Lab-grade peristaltic pumps typically allow stacking multiple pump heads on a single motor, which will get you identical rotation rates and with the same size tubing fairly close flow rates.
If that is not close enough (some fluid property making pumping different? one line pulling air?), you can either go back to closed loop and have a manual differential trim control to run one faster, or you can build a closed loop controller that watches some liquid level sensor, or perhaps a scale under the experiment.