You just need a stateful visitor. I'm typing this from a vague memory of exactly what visitors look like, but you should be able to fix any mistakes I make.
class process_stuff_visitor : public boost::static_visitor<void> {
B* m_b;
public:
process_stuff_visitor(B* b) : m_b(b) {}
void visit(type_a* a) const { m_b->m_var = a->fn1(); }
void visit(type_b* b) const { m_b->m_var = b->fn2(); }
};
// actual processing:
boost::apply_visitor(v[0], process_stuff_visitor(the_b));
Alternatively since you assign to the same member of B, you could just extract the value-generating part.
struct generate_stuff_visitor : public boost::static_visitor<TypeOfMVar> {
TypeOfMVar visit(type_a* a) const { return a->fn1(); }
TypeOfMVar visit(type_b* b) const { return b->fn2(); }
};
the_b->m_var = boost::apply_visitor(v[0], generate_stuff_visitor());