From the wording of your question it sounds like you want the server framework to set the following header for you so that the data flows back in chunked blocks:

Transfer-Encoding: chunked

To do that you need to supply a Spring resource where the full size is not known in advance. You have used ByteArrayResource where the full size is known and results in a Content-Length header being set in the response and chunked is not used.

Change your code to use InputStreamResource and the service will stream the response back to the client with a chunked transfer encoding and no content length. Here's a sample (syntax unchecked):

try(ByteArrayInputStream bis = new ByteArrayInputStream(fileDetailDto.getData())) {
  return ResponseEntity.ok().contentType(new MediaType("text", "csv"))
          .header(HttpHeaders.CONTENT_DISPOSITION,
              "attachment; filename=\"" + fileDetailDto.getFileName() + "\"")
          .body(new InputStreamResource(bis));
}

While this will get you a chunked response I'm not convinced it's the root of your problems with the browser because they are all very capable of asynchronously streaming back data regardless of how the server provides it.

You are assuming program to be downloading whole chunk size, but it may not be the case, because java says:-

An attempt is made to read as many as len(CHUNK_SIZE) bytes, but a smaller number may be read, possibly zero.

This may work:-

byte[] temp = new byte[CHUNK_SIZE];
int bytes = 0;
for(int a = 0; a < chunks; a++) {
    bytes =remoteStream.read(temp, 0, CHUNK_SIZE);
    if(bytes == -1) {
      System.out.println("Downloaded last chunk : Terminating ");
      break;
    }
    stream.write(temp, 0 ,bytes);
}

I am creating a huge byte[] with the size of the online file

Why? You don't need a byte array the size of the file. That just wastes space and adds latency. Just read and write buffers:

while ((count = in.read(buffer)) > 0)
{
    out.write(buffer, 0, count);
}

This works for any byte[] buffer of size greater than zero. I generally use 8192.

With URLConnection you don't need the content-length either: just read to EOS as above.