Move-only version of std::function

No, there is no move-only version of std::function in the C++ std library. (As of C++14)

Fastest possible delegates is an implementation of a std::function like class that happens to be faster than most std::function implementations in many std libraries, and it should be easy to fork into a move and copy version.

Wrapping your move only function object into a shared_ptr<F> in a class with a forwarding operator() is another approach.

Here is a task sketch:

template<class Sig>
struct task;

namespace details {
  template<class Sig>
  struct task_iimpl;
  template<class R, class...Args>
  struct task_iimpl<R(Args...)> {
    virtual ~task_iimpl() {}
    virtual R invoke(Args&&...args) const = 0;
  };
  template<class F, class Sig>
  struct task_impl;
  template<class F, class R, class...Args>
  struct task_impl<F,R(Args...)>:
    task_iimpl<R(Args...)>
  {
    F f;
    template<class T>
    task_impl(T&& t):f(std::forward<T>(t)) {}
    virtual R invoke(Args&&...args) const override {
      return f( std::forward<Args>(args...) );
    }
  };
  template<class F, class...Args>
  struct task_impl<F,void(Args...)>:
    task_iimpl<void(Args...)>
  {
    F f;
    template<class T>
    task_impl(T&& t):f(std::forward<T>(t)) {}
    virtual void invoke(Args&&...args) const override {
      f( std::forward<Args>(args...) );
    }
  };
}
template<class R, class...Args>
struct task<R(Args...)> {
  virtual ~task_iimpl() {}
  R operator()(Args...args) const {
    return pImpl->invoke(std::forward<Args>(args...));
  }
  explicit operator bool()const{ return static_cast<bool>(pImpl); }
  task(task &&)=default;
  task& operator=(task &&)=default;
  task()=default;

  // and now for a mess of constructors
  // the rule is that a task can be constructed from anything
  // callable<R(Args...)>, destroyable, and can be constructed
  // from whatever is passed in.  The callable feature is tested for
  // in addition, if constructed from something convertible to `bool`,
  // then if that test fails we construct an empty task.  This makes us work
  // well with empty std::functions and function pointers and other tasks
  // that are call-compatible, but not exactly the same:
  struct from_func_t {};
  template<class F,
    class dF=std::decay_t<F>,
    class=std::enable_if_t<!std::is_same<dF, task>{}>,
    class FR=decltype(std::declval<F const&>()(std::declval<Args>()...)),
    std::enable_if_t<std::is_same<R, void>{} || std::is_convertible<FR, R>{} >*=0,
    std::enable_if_t<std::is_convertible<dF, bool>{}>*=0
  >
  task(F&& f):
    task(
      static_cast<bool>(f)?
      task( from_func_t{}, std::forward<F>(f) ):
      task()
    )
  {}
  template<class F,
    class dF=std::decay_t<F>,
    class=std::enable_if_t<!std::is_same<dF, task>{}>,
    class FR=decltype(std::declval<F const&>()(std::declval<Args>()...)),
    std::enable_if_t<std::is_same<R, void>{} || std::is_convertible<FR, R>{} >*=0,
    std::enable_if_t<!std::is_convertible<dF, bool>{}>*=0
  >
  task(F&& f):
    task( from_func_t{}, std::forward<F>(f) )
  {}

  task(std::nullptr_t):task() {}
  // overload resolution helper when signatures match exactly:
  task( R(*pf)(Args...) ):
    task( pf?task( from_func_t{}, pf ):task() )
  {}
private:
  template<class F,
    class dF=std::decay_t<F>
  >
  task(from_func_t, F&& f):
    pImpl( std::make_unique<details::task_impl<dF,R(Args...)>>(
      std::forward<F>(f)
    )
  {}

  std::unique_ptr<details::task_iimpl<R(Args...)> pImpl;
};

but it has not been tested or compiled, I just wrote it.

A more industrial strength version would include a small buffer optimization (SBO) to store small callables (assuming they are movable; if not movable, store on heap to allow moving), and a get-pointer-if-you-guess-the-type-right (like std::function).