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Assuming the site-to-site data copy is using TCP, often the issue is TCP receiver isn't supporting a TCP window large enough for the bandwidth delay product. Assuming RTT of 100 ms @ 10 Mbps, you need 125 KB of buffer space. (NB: it also help to insure you can buffer TCP slow start on you network bottlenecks.)
Another issue that commonly impacts performance across VPNs is packet fragmentation. (NB: adhereing to recommendation for avoiding packet recommendation avoid this issue.)
Lastly, when and if working with broadband connections that don't match physical interface bandwidth, providers may police traffic rate in such away that nominal bandwidth capacity isn't obtainable. (NB: this can be avoided by shaping to your contracted bandwidth.)
