Efficient pangenome indexes are promising tools for many applications, including rapid classification of nanopore sequencing reads. Recently, a compressed-index data structure called the "move structure" was proposed as an alternative to other BWT-based indexes like the FM index and r-index. The move structure uniquely achieves both O(r) space and O(1)-time queries, where r is the number of runs in the pangenome BWT. We implemented Movi, an efficient tool for building and querying move-structure pangenome indexes. While the size of the Movi's index is larger than the r-index, it scales at a smaller rate for pangenome references, as its size is exactly proportional to r, the number of runs in the BWT of the reference. Movi can compute sophisticated matching queries needed for classification - such as pseudo-matching lengths and backward search - at least ten times faster than the fastest available methods, and in some cases more than 30-fold faster. Movi achieves this speed by leveraging the move structure's strong locality of reference, incurring close to the minimum possible number of cache misses for queries against large pangenomes. We achieve still further speed improvements by using memory prefetching to attain a degree of latency hiding that would be difficult with other index structures like the r-index. Movi's fast constant-time query loop makes it well suited to real-time applications like adaptive sampling for nanopore sequencing, where decisions must be made in a small and predictable time interval.