Suppose that $G$ and $H$ are locally compact Hausdorff groups.
EDIT: As Eric Wofsey pointed out in the comments; the group structures can probably be ignored for this question.
Assume that for each $x\in G$ we have a compactly supported continuous function $\phi_x\colon H\to\mathbb{C}$. That is, for each $x\in G$ there exist a compact set $L_x$ in $H$ such that $\phi_x(y)=0$ for all $y\notin L_x$.
Also assume that $(x,y)\mapsto\phi_x(y)$ is jointly continuous and that the map $$G\ni x\mapsto\phi_x\in C_c(H)$$ is compactly supported. That is, there exists a compact set $K$ in $G$ such that $\phi_x\equiv0$ (identically zero) whenever $x\notin K$.
Q: Is the map $(x,y)\mapsto\phi_x(y)$ compactly supported?.
If I sketch this for $G=H=\mathbb{R}$, then this seems to be true. In general I believe that $\phi_x(y)=0$ for $$(x,y)\notin K\times\bigcup_{x\in K}L_x,$$ but I do not know if this set is compact. Any suggestions would be greatly appreciated.
This is not true even in the case $G=H=\mathbb{R}$. For instance, consider a continuous function $\phi:\mathbb{R}^2\to\mathbb{R}$ which vanishes outside $\bigcup_{n\in\mathbb{Z}_+}[1/(n+1),1/n]\times[n,n+1]$ and is nonzero somewhere on each one of those rectangles. Writing $\phi_x(y)=\phi(x,y)$, then each $\phi_x$ has compact support and $x\mapsto\phi_x$ has compact support, but $\phi$ itself does not have compact support.