This likely happens because you have several rows in the post-meta table with huge amounts of data, probably dozens of megabytes, maybe even a hundred megabytes per row.
WP Staging is so smart that it considers the maximum allowed memory consumption and reduces the rows until it can continue the copying process.
Once it has copied these huge rows it will speed up again and copy the rest with full speed. You just have to be patient until it passes these rows.
I also recommend that you look into the first rows of the post meta table to see what data it is that big. Usually, such big rows are an indicator of having a problematic plugin.
Please let me know how it goes.