The new crosswalks are not designed for source data for whole block groups. The reason for this design is that (to my knowledge) all 1990 and 2000 census data tables that are available for block groups are also available for blocks and/or block group parts (BGPs), and to standardize geographic units across time, allocating data from the smallest possible source units (blocks or BGPs) will generally be more accurate, as discussed here in the crosswalk documentation.
It would be possible to use the BGP crosswalks to allocate data from block groups, but you’d first have to disaggregate from block groups down to BGPs, following steps similar to those I laid out for using the block crosswalks for the same purpose. Such disaggregation also degrades accuracy.
I’m wondering if for the data you have at the block-group level, could you get the same statistics at the BGP level? That should be possible using NHGIS, following this guidance. If you’re able to get BGP-level data, then you can use the crosswalks as they are without additional manipulation.