When you writing "instrument" you mean research method?

If so, it proposes qualitative research.

First do laboratory experyment checking the behavior of two different groups - intercultural and onecultural.

Next  make Interviews (focus group) with  intercultural and  onecultural groups

This allowed advisable to check your characteristics.  

Write me please what exactly you want to explore - maybe than I can help

A frequently used instrument to measure cultural awareness is the one developed by Bennett, M. J., & Hammer, M. R. (1998). The intercultural development inventory (IDI) manual. Portland, OR: The Intercultural Communication Institute.

The following article gives an overview on several methods: http://www.iaccp.org/sites/default/files/stellenbosch_pdf/Matveev.pdf

and a nice introductory text is given by UNESCO: 

http://unesdoc.unesco.org/images/0021/002197/219768e.pdf

I've just made a brief reading to the articles you suggested. Thank you... I wonder if isn't been created other instruments as laboratory test, or questionaries, or other test.

You could be interested in the "Related work on social interaction and social simulation" that describes different scenarios of intercultural communication in section 2.5.

Chapter Related Work on Social Interaction and Social Simulation

Ethnography in the class room or Phenomenology could be optional methods for your research. 

Cultural competence is the ability to teach students & entails developing certain  interpersonal sensitivities,  certain bodies of cultural knowledge and mastering a set of skills that, taken together, underlie effective cross-cultural teaching.

You can use the M-GRISMS-M was designed and validated to assess cultural bias attitudes toward others, racial; ethnic; tolerance toward other; and sexual prejudices.

 Dear Ernesto,

I would recctomandad several references:

Council of Europe. 2008. Autobiography of Intercultural Encounters. Strasbourg: Council of Europe.

Council of Europe. 2012. Intercultural competence for all. Strasbourg: Council of Europe.

Council of Europe. 2014. Developing intercultural competence through education. Strasbourg: Council of Europe.

Deardorff, D.K. 2015. Demystifying Outcomes Assessment for International Educators. Sterling: Stylus.

Holmes, P. and O’Neill, G. 2010. “Autoethnography and Self.Reflections: Tools for Self-Assessing Intercultural Competence.” In Becoming Intercultural: Inside and Outside the Classroom, edited by J.Tsau and S. Houghton, 167-193. Cambridge: Cambridge Scholars.

Holmes, P. and O’Neill, G. 2012. “Developing and evalutating intercultural competence: Ethnograhies of intercultural encounters.” International Journal of Intercultural Relations 36 (5): 707-718. doi: 10.1016/j.ijintrel.2012.04.010

Best wishes,

Mattia (Italy)

Dear Ernesto,

I think, the field work can be an effective technique for your work... particularly, phenomenology and group works... But, political structures  is also very important...

Thus, you can produce more efficient results with a mix method...

You can check out the following case studies ...

It can contribute to your work...

1- Multiculturality Attitude Scale: A Sample from Turkey.

http://www.iojes.net/userfiles/Article/IOJES_1733.pdf 

2- A critical approach to discipline of human geography as an apparatus for state hegemony in Turkey

http://www.tandfonline.com/doi/full/10.1080/03098265.2015.1116061

Best Wishes

Nurettin

Cultural competence is the ability to develop certain interpersonal sensitivities, cultural knowledge.

Attached you will find a framework that I like.  It is not an instrument, but it might form the basis for developing an instrument.

Data Levels of Cross-Cultural Understanding

The M-GRISMS-M is an excellent instrument to assess the 'ISMs', e.g., racism, sexism, heterosexism, cultural prejudices, religious intolerance, etc.. The instrument has high test-retest reliability and criterion validity.

If I were you, I would design your own technique based on the insights you can get from M-GRIMS-M or the old IDI tool (by Bennett). They both have been critisised and, for me, do not reflect all the complexity of our human behaviour and experience. Therefore I would definitely choose something more based on semi-structured interviews, auto reports or even some alternative projection-like techniques. However, it will, definitely, depend on the scale of sample. Good luck and, please, share the results...