C++ Lambda Code Generation with Init Captures in C++ 14

Case 1 [x](){}: The generated constructor will accept its argument by possibly const-qualified reference to avoid unnecessary copies:

__some_compiler_generated_name(const int& x) : x_{x}{}

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Case 2 [x&](){}: Your assumptions here are correct, x is passed and stored by reference.

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Case 3 [x = 33](){}: Again correct, x is initialized by value.

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Case 4 [p = std::move(unique_ptr_var)]: The constructor will look like this:

    __some_compiler_generated_name(std::unique_ptr<SomeType>&& x) :
        x_{std::move(x)}{}

so yes, the unique_ptr_var is "moved into" the closure. See also Scott Meyer's Item 32 in Effective Modern C++ ("Use init capture to move objects into closures").

There's less of a need to speculate, using cppinsights.io.

Case 1:
Code

#include <memory>

int main() {
    int x = 33;
    auto lambda = [x]() { std::cout << x << std::endl; };
}

Compiler generates

#include <iostream>

int main()
{
  int x = 6;

  class __lambda_5_16
  {
    int x;
    public: 
    inline void operator()() const
    {
      std::cout.operator<<(x).operator<<(std::endl);
    }

    // inline /*constexpr */ __lambda_5_16(const __lambda_5_16 &) = default;
    // inline /*constexpr */ __lambda_5_16(__lambda_5_16 &&) noexcept = default;
    public: __lambda_5_16(int _x)
    : x{_x}
    {}

  };

  __lambda_5_16 lambda = __lambda_5_16(__lambda_5_16{x});
}

Case 2:
Code

#include <iostream>
#include <memory>

int main() {
    int x = 33;
    auto lambda = [&x]() { std::cout << x << std::endl; };
}

Compiler generates

#include <iostream>

int main()
{
  int x = 6;

  class __lambda_5_16
  {
    int & x;
    public: 
    inline void operator()() const
    {
      std::cout.operator<<(x).operator<<(std::endl);
    }

    // inline /*constexpr */ __lambda_5_16(const __lambda_5_16 &) = default;
    // inline /*constexpr */ __lambda_5_16(__lambda_5_16 &&) noexcept = default;
    public: __lambda_5_16(int & _x)
    : x{_x}
    {}

  };

  __lambda_5_16 lambda = __lambda_5_16(__lambda_5_16{x});
}

Case 3:
Code

#include <iostream>

int main() {
    auto lambda = [x = 33]() { std::cout << x << std::endl; };
}

Compiler generates

#include <iostream>

int main()
{

  class __lambda_4_16
  {
    int x;
    public: 
    inline void operator()() const
    {
      std::cout.operator<<(x).operator<<(std::endl);
    }

    // inline /*constexpr */ __lambda_4_16(const __lambda_4_16 &) = default;
    // inline /*constexpr */ __lambda_4_16(__lambda_4_16 &&) noexcept = default;
    public: __lambda_4_16(int _x)
    : x{_x}
    {}

  };

  __lambda_4_16 lambda = __lambda_4_16(__lambda_4_16{33});
}

Case 4 (unofficially):
Code

#include <iostream>
#include <memory>

int main() {
    auto x = std::make_unique<int>(33);
    auto lambda = [x = std::move(x)]() { std::cout << *x << std::endl; };
}

Compiler generates

// EDITED output to minimize horizontal scrolling
#include <iostream>
#include <memory>

int main()
{
  std::unique_ptr<int, std::default_delete<int> > x = 
      std::unique_ptr<int, std::default_delete<int> >(std::make_unique<int>(33));

  class __lambda_6_16
  {
    std::unique_ptr<int, std::default_delete<int> > x;
    public: 
    inline void operator()() const
    {
      std::cout.operator<<(x.operator*()).operator<<(std::endl);
    }

    // inline __lambda_6_16(const __lambda_6_16 &) = delete;
    // inline __lambda_6_16(__lambda_6_16 &&) noexcept = default;
    public: __lambda_6_16(std::unique_ptr<int, std::default_delete<int> > _x)
    : x{_x}
    {}

  };

  __lambda_6_16 lambda = __lambda_6_16(__lambda_6_16{std::unique_ptr<int, 
                                                     std::default_delete<int> >
                                                         (std::move(x))});
}

And I believe this last piece of code answers your question. A move occurs, but not [technically] in the constructor.

Captures themselves aren't const, but you can see that the operator() function is. Naturally, if you need to modify the captures, you mark the lambda as mutable.