To analyze data at the block group level, it can initially seem intuitive to collect input data from multiple years at the block group level. But in fact, as discussed fully on the NHGIS crosswalks page, that’s not necessary, and to generate a time series of data for one year’s block groups, it’s generally more accurate to start with data from the smallest possible units. The smallest geographic areas for which the Census Bureau published 1990 and 2000 long-form census data (including data on household income) are block group parts. For this reason, NHGIS provides crosswalks from 1990 and 2000 block group parts (and not from 1990 and 2000 block groups). This section describes how you can use NHGIS to obtain data for 1990 and 2000 block group parts. You could then link that data to the NHGIS crosswalks to generate data for 2010 block groups.
Another issue is that NHGIS crosswalks are designed to support the standardization of count data, such as the number of households in block groups. See this forum post for guidance on how to handle a non-count statistic like median household income.