Transporter technology is widely debated. Through most of TOS the transporter appears to lock the subjects in stasis—taking a "snapshot" of their state at an instant in time. That subject is then "destroyed" or converted to energy while the snapshot pattern is stored in a computer buffer. When the pattern is projected elsewhere, perhaps like a phaser beam, the subject is reassembled.
On the other hand, one TOS episode, "That Which Survives," shows Kirk witnessing an attack on the transporter operator after de-rez has already begun. A similar thing can be seen in the movie THE WRATH OF KHAN where Kirk and Saavik are holding a conversation during transport, suggesting that the transporter is not
some kind of ultra-fax, but a wormhole-like gateway.
I remember one TNG episode where engineer Montgomery Scott is found in the buffer of a transporter. His ship was in dire trouble, and he came up with a way to save himself through a kind of high-tech "suspended animation." He rigged the buffer to prevent pattern degradation over an extended period.
So, whether or not the different "matter transmitter" systems are compatible is entirely up to the writer of a given story. Oh, and there was, of course, the transporter accident in ST:TMP that seemed like a poorly tuned radio problem. (Olde fashioned whistling and static. "Boost your gain!" and all that.)