Named rvalue references

Please pardon my lack of clarity on this topic. I am trying to create functions for inserting a big class into a vector. In this example, I use vector of ints as the big class.

#include <vector>
#include <iostream>
using namespace std;

vector<vector<int>> vectorOfVectors;

void fn(const vector<int> &v) {
    vectorOfVectors.push_back(v);

}
void fn(vector<int> &&v) {
    vectorOfVectors.push_back(std::move(v));
}

int main() {
    vector<int> a({1});
    const vector<int> b({2});
    fn(std::move(a));
    fn(b);
    cout<<b[0];
}

Obviously, I want no copying to be done when possible. My questions:

1. Does this code do the right thing?
2. Is there a better way of doing this?
3. For the same approach to work with custom classes, do I need to define move constructors?