How to start process improvement in any company?

Like Stéphane proposed, it is similar to what I have done in some organisation. Once the ball get rolling you will have more and more support.

To ingrain a culture of process improvement awareness in your company, it must start from the leadership level and cascaded down the hierarchy. It becomes a mandate and resistance is minimized among the ranks. Process improvemement is continuous.Trainings, meetings must be conducted across the board educating the staff of the benefits and gains of process improvement and how to execute these fine-tuning or streamline of processes.

You can start in your team/department by a walkthrough process wherein, you identify the value adding or the non-value addin steps in your processes not to mention, eliminating waste and improving efficiency.