I see transporting (or at least one method thereof) as "bumping" an object out of reality and slinging it in a direction at several times light speed (momentum in the realities being separate things), hoping that it "falls" back into our dimension correctly.
Beaming an object through something that also exists in that reality would be most... unwise... and slight miscalculations would be disastrous.
You would have to:
have sufficient energy to do this to an object of that mass
know the direction you are aiming at (and have to point a turret at it?)
know how far you have to go
know how fast your particular transporter sends things of that mass
know what duration of jump is necessary, taking the last two into account
lead your target because of the time it takes to cycle the transporter
fire the transporter and hope the object gets completely converted before it is launched
hope nothing interferes with it on the way ('natural' navigational hazards in the reality you shot through, or other reality-hopping tech)
hope it lands in gas or vacuum instead of telefragging
hope it even "lands" fully in our dimension (no pieces lost, etc.)
hope it is not harmed by its infinitesimal period in a fundamentally different reality. (didn't spontaneously transmute or distort, etc.)
This form of transporting is basically an external jumpdrive, and would probably be very difficult, requiring fast and well-programmed computers and possibly a skilled operator.
The simplest form of transporter would convert an object into some kind of energy field that can be confined, accelerated, and transmitted like a bolt from weapons/received with antennas; accepted by a specialized replicator as pattern and supplies.
Good for fast transport, bad for going where you are unwelcome (no beaming to a non-transporter target, an interval is allowed for the computer to accept/reject a pattern with basic information [signal strength, mass and, at higher tech levels, composition?]).
I can also see crafty players using these fields as a form of energy that is easy to transfer, using specialized devices to "swallow" the pattern and discard the structure, outputting some form of usable energy instead of the burst of HV static the base pattern is.
Superconductors would probably be necessary for this kind of transporter, as every amp you lose you have to replace, and the amount of energy in a pattern being astronomical for anything of significant mass/complexity.
A spread on the field when projected through space would both help aiming and increase power draw for long beamings: catching 20% of the field means you might be able to reconstruct it, but unless it was boosted at the source you need to supply the other 80% of energy. E=mc^2, so that will be painful. Such a weak signal, unboosted at the source, might still arrive corrupted anyway!
Moral: Put a good amplifier and power supply on your transporter if possible, and try not to beam too far. The receiving party can only compensate for a weak signal so much. Beaming from or to a man-portable, battery-powered system over any distance would probably result in injury if the other party didn't boost the signal. Small antenna = less of you arrives, small power supply = less replaced; unboosted transmitter = only 100% signal sent in the first place.