Construction Grammar (CxG) has recently been used as the basis for probing studies that have investigated the performance of large pretrained language models (PLMs) with respect to the structure and meaning of constructions. In this position paper, we make suggestions for the continuation and augmentation of this line of research. We look at probing methodology that was not designed with CxG in mind, as well as probing methodology that was designed for specific constructions. We analyse selected previous work in detail, and provide our view of the most important challenges and research questions that this promising new field faces.