This paper presents an extension of the results of [l] to age-dependent branching processes. At the same time (and without much additional difficulty) we deal with a branching process on a general state space, rather than with the special model of the binary cascade treated in [l]. We start with a “standard” age-dependent process, defined as in chapter VI of Harris [2]. An initial “parent” particle splits after a random time T into a random number of “offspring” particles. Each of the offspring acts independently as a parent, and after random times (independently distributed as 2’) produce the next generation of offspring, etc. Let N, denote the number of particles existing at time t. The process to be studied here is constructed from the standard one by associating with each particle a “ type,” namely, a point x in a d-dimensional Euclidian space Q. Thus at any given time, each particle existing at that time is to be considered as located at a point in 9. (In various applications the coordinates of x will be such quantities as the energy, size, age, location of the associated particle.) Our purpose is to study the diffusion of the particles throughout Q. Let A be a subset of Q, N,(A) be the number of particles in A at time t, and M,(A) = N,(A)/N, be the proportion of particles in A at t. Note that M,(e) is a random measure; i.e., for each sample path (realization) of the branching process, AM,(.) is a measure for each t. To obtain a nondegenerate limit law we shall let the set A vary with time, and consider a process of the form M,(A,). We shall show that by letting iI, grow in a suitable manner, we can attain the convergence (in mean square) of &‘,(A,) to a Gaussian probability function. This, essentially, is the content of Theorem 3 and Remark 3 below.