If you use DriveHQ FileManager, Online Backup or WebDAV Drive Mapping to transfer files, the file modification time is preserved. However, the file creation time is changed to the current time.
If you use a web browser to upload/download files, both file modification time and creation time are changed to the current time.
If you use FTP to transfer files, it is dependent on your FTP client software. With FileZilla FTP client, you can change the option to keep the file modification time or not. Please see the screenshot below: (By default, it keeps the modification time. You need to uncheck the checkbox. The file creation time is always changed to the current time.)