  We prove existence of variational solutions for a class of doubly nonlinear
nonlocal evolution equations whose prototype is the double phase equation
\begin{align*} \partial_t u^m &+ \text{P.V.}\int_{\mathbb{R}^N}
\frac{|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{N+ps}}\\&+a(x,y)\frac{|u(x,t)-u(y,t)|^{q-2}(u(x,t)-u(y,t))}{|x-y|^{N+qr}}
\,dy = 0,\,m>0,\,p>1,\,s,r\in (0,1). \end{align*}
  We make use of the approach of minimizing movements pioneered by DeGiorgi and
Ambrosio and refined by B\"ogelein, Duzaar, Marcellini, and co-authors to study
nonlinear parabolic equations with non-standard growth.
