Thanks Robert,

Is there any way to achieve this with workflow? I've tried the replicate entries activity, but that doesn't seem to do the trick?

Use a script activity to export the entry as a briefcase to a temporary file, then import the briefcase:

string tempPath = Path.GetTempFileName();

try
{
    using (BriefcaseExporter briefcaseExporter = new BriefcaseExporter(session))
    {
        briefcaseExporter.BriefcaseName = "briefcase.lfb";
        briefcaseExporter.IncludeVersionHistory = true;
        briefcaseExporter.AddEntry(docId);
        briefcaseExporter.Export(tempPath);
    }

    using (BriefcaseImporter importer = new BriefcaseImporter(session))
    {
        importer.DestinationFolder = Folder.GetFolderInfo(@"\DestinationFolder", session);
        importer.OnDocumentConflict = DocumentConflictStrategy.MakeUnique;

        importer.Import(tempPath);
        Exception ex = importer.GetLastException();
        if (ex != null)
            throw ex;
    }
}
finally
{
    // Cleanup temp file
    File.Delete(tempPath);
}

Did you check "include version history" in the "copy settings" dialog in Replicate Entries?

Thanks Miruna, this does replicate the entry with the version history (although it does create a new mystery version for some reason). 

The last bit I'm struggling with is how to get to the metadata of the replicated entry, as I need to tidy up the meta data for the multi value field (to only include the current 'for each value' loop) and to clear the other values in the multi value field.

I can only seem to interact with folders, not the copied entry itself?

In a move entry activity I get options for the entry itself:-

Any ideas?