  We study a class of dynamically constructed point processes in which at every
step a new point (particle) is added to the current configuration with a
distribution depending on the local structure around a uniformly chosen
particle. This class covers, in particular, generalised Polya urn scheme,
Dubbins--Freedman random measures and cooperative sequential adsorption models
studied previously. Specifically, we address models where the distribution of a
newly added particle is determined by the distance to the closest particle from
the chosen one. We address boundedness of the processes and convergence
properties of the corresponding sample measure. We show that in general the
limiting measure is random when exists and that this is the case for a wide
class of almost surely bounded processes.
