Let X = G/K be a symmetric space of noncompact type, L be the Laplace-Beltrami operator on X, and b the bottom of its spectrum. In this paper we study the behaviour of the Lp-Lq operator norms of the (σ,θ)-Poisson semigroup as t tends to zero and to infinity for all such p and q. As t tends to infinity the behaviour of these norms involves power of t in which the rank (r) and the pseudo-dimension of X play an important role. Two features of our study are noteworthy. First, while the θ-heat semigroup is Lp-Lq bounded whenever 1≤p≤q≤∞, the (σ,θ)-Poisson semigroup is not Lp-Lq bounded for many such p and q. Second, when p and q reach the critical index for Lp-Lq boundedness, the exponent r+1 appears.