For bounded variable exponents \(p(\cdot):\mathbb{R}^n\to[1,\infty)\) and \(q(\cdot):\mathbb{R}_+\to[1,\infty)\), consider the corresponding variable exponent Lebesgue spaces \(L^{p(\cdot)}(\mathbb{R}^n)\) and \(L^{q(\cdot)}(\mathbb{R}_+,\frac{dt}{t})\), respectively. For \(0<\gamma<\delta\) and a suitable function \(\omega:\mathbb{R}^n\times\mathbb{R}_+\to\mathbb{R}_+\), the authors introduce the global variable exponent Herz space \(H_{\omega(\cdot,\cdot),\gamma,\delta}^{p(\cdot),q(\cdot)}(\mathbb{R}^n)\) as the set of functions \(f\) on \(\mathbb{R}^n\) such that \[ \|f\|:=\sup_{\xi\in\mathbb{R}^n} \left\| \omega(\xi,t)\left\| f\chi_{B(\xi,\delta t)\setminus B(\xi,\gamma t)} \right\|_{L^{p(\cdot)}(\mathbb{R}^n)} \right\|_{L^{p(\cdot)}(\mathbb{R}_+,\frac{dt}{t})}<\infty. \] Similarly, for a suitable function \(\omega:\mathbb{R}_+\to\mathbb{R}_+\), the local variable exponent Herz space is defined as the set of functions \(f\) on \(\mathbb{R}^n\) such that \[ \|f\|:= \left\| \omega(t)\left\| f\chi_{B(0,\delta t)\setminus B(0,\gamma t)} \right\|_{L^{p(\cdot)}(\mathbb{R}^n)} \right\|_{L^{p(\cdot)}(\mathbb{R}_+,\frac{dt}{t})}<\infty. \] Here \(B(x,r)\) denotes the ball centered at \(x\in\mathbb{R}^n\) of radius \(r>0\). The authors show that variable Morrey type spaces and complementary variable Morrey type spaces are included into the scale of these generalized variable Herz spaces. They also prove the boundedness of a class of sublinear operators \(T\), satisfying the size condition \(|Tf(x)|\le C\int_{\mathbb{R}^n}\frac{|f(y)|}{|x-y|^n}dy\) for \(x\notin\mathrm{supp}\,f\), in generalized variable Herz spaces with application to variable Morrey type spaces and their complementary spaces, based on the mentioned inclusion.