diff --git a/fs/operations/multithread.go b/fs/operations/multithread.go
index e5357dc0b..d9e83be42 100644
--- a/fs/operations/multithread.go
+++ b/fs/operations/multithread.go
@@ -11,7 +11,6 @@ import (
 	"github.com/rclone/rclone/fs/accounting"
 	"github.com/rclone/rclone/lib/atexit"
 	"github.com/rclone/rclone/lib/multipart"
-	"github.com/rclone/rclone/lib/readers"
 	"golang.org/x/sync/errgroup"
 )
 
@@ -53,13 +52,13 @@ func doMultiThreadCopy(ctx context.Context, f fs.Fs, src fs.Object) bool {
 
 // state for a multi-thread copy
 type multiThreadCopyState struct {
-	ctx       context.Context
-	partSize  int64
-	size      int64
-	src       fs.Object
-	acc       *accounting.Account
-	numChunks int
-	noSeek    bool // set if sure the receiving fs won't seek the input
+	ctx         context.Context
+	partSize    int64
+	size        int64
+	src         fs.Object
+	acc         *accounting.Account
+	numChunks   int
+	noBuffering bool // set to read the input without buffering
 }
 
 // Copy a single chunk into place
@@ -88,10 +87,11 @@ func (mc *multiThreadCopyState) copyChunk(ctx context.Context, chunk int, writer
 	defer fs.CheckClose(rc, &err)
 
 	var rs io.ReadSeeker
-	if mc.noSeek {
+	if mc.noBuffering {
 		// Read directly if we are sure we aren't going to seek
 		// and account with accounting
-		rs = readers.NoSeeker{Reader: mc.acc.WrapStream(rc)}
+		rc.SetAccounting(mc.acc.AccountRead)
+		rs = rc
 	} else {
 		// Read the chunk into buffered reader
 		rw := multipart.NewRW()
@@ -130,15 +130,25 @@ func calculateNumChunks(size int64, chunkSize int64) int {
 func multiThreadCopy(ctx context.Context, f fs.Fs, remote string, src fs.Object, concurrency int, tr *accounting.Transfer, options ...fs.OpenOption) (newDst fs.Object, err error) {
 	openChunkWriter := f.Features().OpenChunkWriter
 	ci := fs.GetConfig(ctx)
-	noseek := false
+	noBuffering := false
 	if openChunkWriter == nil {
 		openWriterAt := f.Features().OpenWriterAt
 		if openWriterAt == nil {
 			return nil, errors.New("multi-thread copy: neither OpenChunkWriter nor OpenWriterAt supported")
 		}
 		openChunkWriter = openChunkWriterFromOpenWriterAt(openWriterAt, int64(ci.MultiThreadChunkSize), int64(ci.MultiThreadWriteBufferSize), f)
-		// We don't seek the chunks with OpenWriterAt
-		noseek = true
+		// If we are using OpenWriterAt we don't seek the chunks so don't need to buffer
+		fs.Debugf(src, "multi-thread copy: disabling buffering because destination uses OpenWriterAt")
+		noBuffering = true
+	} else if src.Fs().Features().IsLocal {
+		// If the source fs is local we don't need to buffer
+		fs.Debugf(src, "multi-thread copy: disabling buffering because source is local disk")
+		noBuffering = true
+	} else if f.Features().ChunkWriterDoesntSeek {
+		// If the destination Fs promises not to seek its chunks
+		// (except for retries) then we don't need buffering.
+		fs.Debugf(src, "multi-thread copy: disabling buffering because destination has set ChunkWriterDoesntSeek")
+		noBuffering = true
 	}
 
 	if src.Size() < 0 {
@@ -192,12 +202,12 @@ func multiThreadCopy(ctx context.Context, f fs.Fs, remote string, src fs.Object,
 	g.SetLimit(concurrency)
 
 	mc := &multiThreadCopyState{
-		ctx:       gCtx,
-		size:      src.Size(),
-		src:       src,
-		partSize:  info.ChunkSize,
-		numChunks: numChunks,
-		noSeek:    noseek,
+		ctx:         gCtx,
+		size:        src.Size(),
+		src:         src,
+		partSize:    info.ChunkSize,
+		numChunks:   numChunks,
+		noBuffering: noBuffering,
 	}
 
 	// Make accounting