Which system is running on your Mac? The Live Photos may be using the new HEIC/HEVC codecs, and only macOS 10.13.x can read this format. If your Mac has not been upgraded to High Sierra, consider to upgrade or set the iPhone to download the photos and videos in a more compatible format.
See: Using HEIF or HEVC media on Apple devices - Apple Support
When you import HEIF or HEVC media from an attached iOS device to Photos, Image Capture, or a PC, the media might be converted to JPEG or H.264.
You can change this import behavior in iOS 11. Go to Settings > Photos. In the "Transfer to Mac or PC" section, tap Keep Originals to prevent the media from being converted when importing.