jmc

Note the prime factorizations $700=2^2\cdot 5^2\cdot 7$ and $7000=2^3\cdot 5^3\cdot 7$.

If $\mathrm{lcm}[r,700]=7000$, then in particular, $r$ is a divisor of $7000$, so we can write $r=2^{\alpha }\cdot 5^{\beta }\cdot 7^{\gamma }$, where $0\le \alpha \le 3$, $0\le \beta \le 3$, and $0\le \gamma \le 1$.

Moreover, we know that $\mathrm{lcm}[r,700]=2^{\max \{\alpha ,2\}}\cdot 5^{\max \{\beta ,2\}}\cdot 7^{\max \{\gamma ,1\}}$, and we know that this is equal to $7000=2^3\cdot 5^3\cdot 7$. This is possible only if $\alpha =3$ and $\beta =3$, but $\gamma$ can be $0$ or $1$, giving us two choices for $r$: $$r=2^3\cdot 5^3\cdot 7^0=1000\text{ or }r=2^3\cdot 5^3\cdot 7^1=7000.$$So the sum of all solutions is $1000+7000=\boxed{8000}$.