Is it prescribed to perform the prostration of forgetfulness (sujood al-sahw) for someone who forgot the takbeer of movement when following the imam in prostration in the last rak'ah, knowing that this rak'ah will not be counted for him?

If you find the Imam prostrating, only the opening Takbir (Takbirat al-Ihram) is sufficient for you, and no other Takbir is required for the transition. Therefore, you are not prescribed to perform the prostration of forgetfulness (Sujud al-Sahw) before the salutation because its cause has not occurred.