Integral of power of binomials equal to sum of power of binomials?

I don't know if there is a relation for values of $\ell$ apart from $1$ and $2$ (that would be very interesting, and surprising to me), but here is a unified way to look at what's going on for exponents $1$ and $2$.

Consider the function on ${\Bbb R}$ defined by $$ f(x) = (1+e^{2\pi i x})^n $$ for $-1/2 \le x \le 1/2$ and $f(x) = 0$ if $|x| >1/2$. We can compute its Fourier transform: $$ {\widehat f}(\xi) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i x\xi} dx, $$ and this turns out to be $$ \binom{n}{\xi} = \frac{n!}{\Gamma(\xi+1) \Gamma(n-\xi + 1)}. $$ (Check for $n=1$, and then use $\binom{n}{\xi} = \binom{n-1}{\xi} + \binom{n-1}{\xi -1}$). So now Fourier inversion gives $$ f(0) = 2^n = \int_{-\infty}^{\infty} \binom{n}{\xi} d\xi, $$ which is the case $\ell =1$.

Next Plancherel gives $$ \int_{-1/2}^{1/2} |f(x)|^2 dx = \int_{-\infty}^{\infty} \binom{n}{\xi}^2 d\xi. $$ Now instead of $f$ consider the $1$-periodic function $F$ defined by $F(x) = f(x)$ on $(-1/2,1/2)$ and extended periodically. The Fourier coefficients of $F$ are $$ {\widehat F}(k) = \int_{-1/2}^{1/2} f(x) e^{-2\pi ik} dx = \binom{n}{k}, $$ and so Parseval gives $$ \sum_{k} {\widehat F}(k)^2 =\sum_{k=0}^{n} \binom{n}{k}^2 = \int_{-1/2}^{1/2} f(x)^2 dx. $$ This gives the $\ell=2$ identity.

The generalization looks like this $$ \int_{-\infty}^{\infty} \binom{n}{\alpha x}^l dx =\sum_{k=-\infty}^\infty\binom{n}{\alpha k}^l,\quad 0<\alpha\le 2/l,~l\in\mathbb{N}\tag{1} $$ where $n$ need not be an integer. The general theorem is given for example in the paper Surprising sinc sums and integrals. Below I give the general outline of the proof which is based on the well known fact that the following function is band limited (its Fourier transform has limited spectrum) $$g(x)=\binom{n}{x}=\frac{1}{2\pi}\int_{-\pi}^{\pi} (1+e^{i \omega})^n e^{- ix\omega} d\omega$$ One can see that Fourier transform is limited to frequencies $|\omega|<\pi$. Whenever spectrum of a function $f(x)$ is limited to frequencies $|\omega|<2\pi$ one expects that $$ \int_{-\infty}^{\infty} f(x) dx =\sum_{k=-\infty}^\infty f(k). $$

Now the Fourier transform of $g(\alpha x)^l$ has a spectrum limited in the band $|\omega|<\pi\alpha l$. This is easy to see calculating Fourier transform $$ \int_{-\infty}^{\infty}g(\alpha x)^le^{-ikx}dx $$ with the help of $\int_{-\infty}^{\infty}e^{-ikx}dx=2\pi\delta(k)$, where $\delta$ is delta function. The general theorem from the paper cited above now states that $$ \int_{-\infty}^{\infty} g(\alpha x)^l dx =\sum_{k=-\infty}^\infty g(\alpha k)^l,\quad 0<\pi\alpha l\le 2\pi,~l\in\mathbb{N}, $$ which is equivalent to (1).

All this analysis also explains that when $\alpha=1$ the proposed identity holds for $l=1,2$ but not for larger $l\in\mathbb{N}$.