We tackle the problems of computing the rightmost variant of the Lempel-Ziv factorizations in the online/sliding model. Previous best bounds for this problem are O(n log n) time with O(n) space, due to Amir et al. [IPL 2002] for the online model, and due to Larsson [CPM 2014] for the sliding model. In this paper, we present faster O(n log n/log log n)-time solutions to both of the online/sliding models. Our algorithms are built on a simple data structure named BP-linked trees, and on a slightly improved version of the range minimum/maximum query (RmQ/RMQ) data structure on a dynamic list of integers. We also present other applications of our algorithms.