No, it's the other way around. Chlorine usage/consumption is a net acidic process. So the initial addition of Trichlor lowers the pH and the usage/consumption of chlorine lowers it even more.
What you describe where the pH changes upon addition of chlorine and returns to normal after the FC has dropped is what happens when you use a hypochlorite source of chlorine such as chlorinating liquid, bleach, Cal-Hypo or lithium hypochlorite. The pH rises upon addition (i.e. when the FC rises) and drops back down upon chlorine usage/consumption (i.e. when the FC drops).
If the TA is on the higher side, then it is true that the pH will try to rise and as the pH drops more carbon dioxide outgassing will occur. So the overall pH drop you see may not be as high as I described. In effect, it is like the TA lowering procedure since adding Trichlor is like adding a combination of chlorine and acid so if the pH does eventually come back up to where it started (without you adding additional pH adjusting chemicals), you would find that the TA was lowered. For every 10 ppm FC added by Trichlor, it lowers the TA by 7 ppm, again this is after the chlorine you added has been used/consumed since the lowering of TA actually occurs during this chlorine usage/consumption. Perhaps the following chart may be helpful to you in understanding what goes on:
.......................... pH . TA . CYA . CH
Trichlor added .... - .... 0 .... + .... 0
Chlorine used ..... - .... -- ... 0 .... 0
------------------------------
Net effect .......... -- ... -- ... + .... 0
.......................... pH . TA . CYA . CH
Dichlor added ..... 0 ... + ... ++ ... 0
Chlorine used ..... - ... -- .... 0 .... 0
------------------------------
Net effect ............ - .... - ... ++ ... 0
.......................... pH . TA . CYA . CH
Cal-Hypo added . + .. ++ ... 0 .... +
Chlorine used ..... - ... -- .... 0 .... 0
------------------------------
Net effect ........... 0 .... 0 ..... 0 .... +
.......................... pH . TA . CYA . CH
Bleach added ...... + .. ++ ... 0 .... 0
Chlorine used ..... - ... -- .... 0 .... 0
------------------------------
Net effect ........... 0 .... 0 ..... 0 .... 0
In the above I am ignoring the small amount of excess lye in the hypochlorite sources of chlorine. Chlorinating liquid and lithium hypochlorite would be the same as bleach above.
Now if you keep the TA higher so that there is more carbon dioxide outgassing, say because you are using Trichlor pucks/tabs on a regular basis, then you have the following where I assume the TA is high enough that the outgassing is enough to keep the pH constant (not usually the case, but useful to show):
.......................... pH . TA . CYA . CH
Trichlor added .... - .... 0 .... + .... 0
Chlorine used ..... - .... -- ... 0 .... 0
CO2 outgassing . ++ .. 0 .... 0 .... 0
------------------------------
Net effect ........... 0 .... -- ... + .... 0
In this extreme case, the pH is stable, but as you can see the TA is dropping so baking soda would need to be added. In practice, one usually has both the pH and TA drop and there is some carbon dioxide outgassing so pH Up is added instead:
.......................... pH . TA . CYA . CH
Trichlor added .... - .... 0 .... + .... 0
Chlorine used ..... - .... -- ... 0 .... 0
CO2 outgassing . + .... 0 .... 0 .... 0
------------------------------
Net effect ........... - .... -- ... + .... 0
With just the precise right amount of outgassing, one can balance everything using pH Up. Usually the TA level where this happens is higher than the 120 ppm normally recommended for Trichlor.