can anyone point towards a solution or method to generate a function that describes the mutation of a function to another over a period of time?
For instance, let $f(x)$ be $x^3$ and $g(x)$ be $x^4$
Over a timeinterval, $dt$, $f(x)$ changes towards $g(x)$ and thus gradually goes from $x^3$ to $x^4$.
Can this be represented in any fashion using mathematics?
I don't really know what you mean by "over a timeinterval, $dt$", so I'm just going to ignore that part of your question.
You could define a function $F(x,t)$ such that $F(x,0)=f(x)$ and $F(x,1)=g(x)$ for all $x$.
There are infinitely many such functions, even if you restrict yourself to continuous ones. Probably the simplest (probably) is $$F(x,t)=(1-t)\cdot f(x) + t\cdot g(x).$$