Well the concept of a constant current is starting to make a bit of sense. A power supply that will increase its internal resistance significantly to the point where the path of least resistance would be the load.
There is no "path of least resistance". This is a concept which simply doesn't apply to electronics. (Ok, there may be a path of least resistance, but not all the current flows through that path).
But what happens to the voltage? It increases?
Let's consider both the theoretical and a practical constant current device.
In a theoretical device the voltage does exactly what is required to maintain a certain current flowing through the device. The voltage may rise, it may fall, it may even change polarity.
In a practical device, there are some limitations. For example, the output voltage will have some defined practical maximum limit. If you need a higher voltage than that, it's simply not going to happen and the current will fall. It is quite common that the voltage can fall no further than zero (very few constant current devices can have an output which changes polarity whilst still delivering a particular current). If you hit this limitation the current may rise without control until some other limit is reached.
Because, what makes sense to me is that more voltage is applied and current stays the same P = IV, that means more power is applied to the load?
It depends on the load. If it is inductive, or if it is storing power in some other way, that power may be delivered back to the current source later. When this happens, it is no longer "real" power" since when averaged ocer a cycle there are positive and negative excursions which cancel out (or nearly cancel out). See
here for more information. Whilst the power is not "real" in that it's not consumed, it is very real to the source of that power which may have unusual peak demands and even the need to absorb power in parts of the waveform.
This applies directly to magnetic fields because if you have a varying magnetic field impinging on a conductor, there will be a change in one direction as the magnetic field grows, which will possibly peak, stop and return to zero as the field becomes static.. Then as the magnetic field is removed, the process repeats in the opposite direction. Over a cycle, the net effect may be zero. (The effect won't be zero if you've taken advantage of it and extracted power or injected power -- in either case you have transferred force from or through the magnetic field, typically from or to whatever is causing the change in the field. Practically this means getting power from the shaft of a generator or applying power to the shaft of a motor)