The purpose of this article is to give a rather thorough understanding of the compact support property for measure-valued processes corresponding to semi-linear equations of the form u(t) = Lu + beta u - alpha u(p) in R-d x (0, infinity), p is an element of (1, 2]; u(x, 0) = f(x) in R-d; u (x, t) >= 0 in R-d x [0, infinity). In particular, we shall investigate how the interplay between the underlying motion (the diffusion process corresponding to L) and the branching affects the compact support property. In [J. Englander, R. Pinsky, On the construction and support properties of measure-valued diffusions on D subset of R-d with spatially dependent branching, Ann. Probab. 27 (1999) 684-730], the compact support property was shown to be equivalent to a certain analytic criterion concerning uniqueness of the Cauchy problem for the semi-linear parabolic equation related to the measured valued process. In a subsequent paper [J. Englander, R. Pinsky, Uniqueness/nonuniqueness for nonnegative solutions of second-order parabolic equations of the form u(t) = Lu + Vu - gamma u(p) in R-n, J. Differential Equations 192 (2003) 396-428], this analytic property was investigated purely from the point of view of partial differential equations. Some of the results obtained in this latter paper yield interesting results concerning the compact support property. In this paper, the results from [J. Englander, R. Pinsky, Uniqueness/nonuniqueness, for nonnegative solutions of second-order parabolic equations of the form u(t) = Lu + Vu - gamma u(p) in R-n, J. Differential Equations 192 (2003) 396-428] that are relevant to the compact support property are presented, sometimes with extensions. These results are interwoven with new results and some informal heuristics. Taken together, they yield a rather comprehensive picture of the compact support property. Inter alia, we show that the concept of a measure-valued process hitting a point can be investigated via the compact support property, and suggest an alternate proof of a result concerning the hitting of points by super-Brownian motion. (c) 2005 Elsevier SAS. All rights reserved.