Photoshop's gradient (or equivalently "Blend" in Ai) is already presented as a way to get a linear color transition between the starting and ending colors. How do we get something nonlinear (a curved transition path) without programming some scripting code?
One solution is to modify the gradient transition path by adjustment layers or blending modes or both. This really produces a continuous transform from the start color to the end color, if the modifying layers have a layer mask that is continuous and makes the affection = Zero at the start and in the end.
Here is a quite complex twist. Layer "Target alone" has the gradient and layer "Modifier" produces a continuous color shift by blending mode "Color" To make all twisty, the modifier itself is a gradient.
More twist is generated by an adjustment layer "Curves" that add contrast. It has the same layer mask as the Modifier, but that's not obligatory. Any mask is ok, if it's continuous and black at the start and in the end.
There exists some uncontinuous blending modes and curves that cause an abrupt jump. One of those is "Hue". Another possiblity to spoil the effect is "no change", a visible constant color area caused by clipping to all zero or all 255. I have not tested or calculated, how much it in practice helps to use 48 bit/pixel RGB mode.
