This paper deals with the large-time behaviour of solutions to the exterior problem of the non-Newtonian filtration equation with first-order term and nonlinear boundary source. In particular, the critical global exponent and the critical Fujita exponent are determined or estimated. An interesting phenomenon is shown: there exists a threshold value for the coefficient of the first-order term such that the critical global exponent is strictly less than the critical Fujita exponent when the coefficient is under this threshold, while these two exponents are identically equal when the coefficient is over this value.