As defined here, the three classes of Markov processes mentioned in the title of this paper have in common the fact that the basic state space is a subset of the reals, and the random trajectories do not jump over points in the state space. They are also regular in that a process starting at any point. in the state space (with the possible exception of the left and right end points), can, with positive probability, reach any other point in the state space. In the discrete-parameter case, any such process has a discrete state space and is a random walk. In the continuous-parameter case, if the state space is an interval, the path functions are continuous, and the process is a diffusion process; if the state space is discrete, the process is a birth and death process. We include both possibilities by allowing the state space to be any closed (except possibly at end points) subset of the reals. These processes are all very similar in their analytic and probabilistic structure. When put in their "natural scale", they are determined by a speed mesure m(dx) nd killing mesure/c (dx) (nd time unit 0 in the case of random walks). It is fairly obvious that in some sense the processes depend continuously on m(dx) and/c(dx). The purpose of this paper is to investigate some of the probabilistic aspects of this continuity. In order to do so, we use a method of It6 and McKean [1] to construct all these processes on a single probability space. This construction involves the use of "local time" for Brownian motion, whose existence and continuity properties were obtained by Trotter [9]. The construction shows that all the processes have local times. If the state space is discrete, the local time is simply the (normalized) occupation time. In Section 1 we summarize the construction in the continuous-parameter case, and in Section 3 we extend it to the discrete-parameter or raudom-walk case. In Section 2 we consider a sequence X(x, ;t), n >= 0, of continuouspmmeter processes with measures m(dx) and ]c(dx) and initiul state x. We suppose that m(dx), (dx), nd x converge suitably to mo(dx), ]co(dx), and x0 respectively, nd that certain other conditions re stisfied. It then follows that several classes of functionals of the X(x ;t) process converge with probability 1 to the corresponding functional of the Xo(xo;t) process.