How does the Thomas-Kilmann Conflict Mode Instrument relate to the Big Five?
There are a few studies that have correlated Big 5 with conflict styles. There are various terminology and measures but many use a similar set of five styles to the TKI.

Antonioni (1998) is highly cited. Here are the correlations.



In short: 


The five conflict styles are correlated in ways that partially reflect the underlying two-dimensions of interested in self and interested in others.
Agreeableness correlates positively with interested in others and avoiding conflict. Agreeableness is also the strongest correlate of conflict styles.
Extraversion and openness are both related to integrating and negatively related to avoiding conflict.
Conscientiousness is related to integrating.
Neuroticism is related to avoiding conflict.


Antonioni, D. (1998). Relationship between the big five personality factors and conflict management styles. International journal of conflict management, 9(4), 336-355.