In this paper, we investigate the stability of some essential spectra of a 2 × 2 block operator matrices pencil with unbounded entries and with non-diagonal domain (i.e a domain consisting of vectors which satisfy certain relations between their components) by using the resolvent of this kind of matrix operator in terms of the union of the essential spectra of the restriction of its diagonal operators entries. Furthermore, an example of two-group transport operators pencils is presented to illustrate the validity of the main results