Many real-world dynamic problems have constraints, and in certain cases not only the objective function changes over time, but also the constraints. However, there is no research in answering the question of whether current algorithms work well on continuous dynamic constrained optimization problems (DCOPs), nor is there any benchmark problem that reflects the common characteristics of continuous DCOPs. This paper contributes to the task of closing this gap. We will present some investigations on the characteristics that might make DCOPs difficult to solve by some existing dynamic optimization (DO) and constraint handling (CH) algorithms. We will then introduce a set of benchmark problems with these characteristics and test several representative DO and CH strategies on these problems. The results confirm that DCOPs do have special characteristics that can significantly affect algorithm performance. The results also reveal some interesting observations where the presence or combination of different types of dynamics and constraints can make the problems easier to solve for certain types of algorithms. Based on the analyses of the results, a list of potential requirements that an algorithm should meet to solve DCOPs effectively will be proposed.