Recent progress has been made in the field of multi-label stream classification, where an instance can be associated with several labels simultaneously. Most recent research has focused on adapting models to the dynamic distribution of non-stationary data streams. However, continual learning is not reduced to adaptation to concept drift: phenomena such as catastrophic forgetting, forward and backward transfers appear when new classification tasks appear in the data stream. The aim of this article is to develop a standardized evaluation protocol specifically adapted to the study of these phenomena, in order to identify the most promising strategies for this new problem of multi-label, multi-task learning on tabular data in a stream. This protocol includes (i) the creation of multi-label and multi-task streams and (ii) an evaluation protocol to measure (a) online performance, (b) phenomena related to continual learning and (c) resource consumption. This protocol is used to compare 12 continual multi-label classification strategies on 4 open literature datasets and 3 simulated datasets. This exploratory analysis has enabled us to identify the promising nature of frugal neural networks coupled with data replay.