Does a constant current source supply constant current even with voltage changes?

I found that constant current sources have a large internal resistance compared to the load resistance which keeps the current constant despite changes in the load resistance. Will this constant current be maintained with voltage changes? And if yes, how?

Shortly put, an ideal current source, if that's what you're considering, will do whatever it takes to maintain the specified constant current. This includes changing its internal resistance to whatever it needs to be.

Edit: in small signal analysis it is considered to be equivalent to open circuit, i.e., very high impedance. It still will do whatever it takes to maintain a constant current, however, regardless what the voltage over it is, and on a large-signal scale resistance will simply be given by Ohm's law.

It is hard to be more specific when the question is not.

The $$vi$$ characteristic of an ideal current source is shown below.

Yiu will note that current stays constant and the terminal potential difference varies as necessary.

A current supply contains circuitry that measures the current flowing through it, compares that to the current setting that you have selected, and then automatically adjusts the voltage output of the supply to achieve and maintain that setpoint.

Since it is expensive and difficult to make a single voltage source that operates at anything from a millivolt to a megavolt, a current source will be limited in the range of voltages it can assert during its operation, and it will "limit out" when pushed outside its specified operating range.

A cheap alternative to an "active" current source like this is to place a resistance in series with your load connected to a voltage source, where the value of this so-called ballast resistor is chosen to be equal to that of the load.

You can show mathematically that this ballast resistor circuit arrangement will hold the current through the load almost constant for small changes in the load resistance.