The answer is complex as it depends on A/C type and engine type as well as operator-specified options.
On our B744s (major carrier that flies into your airport) we have three ratings for takeoff (TO, TO1 and TO2) as well as three climb (CLB) ratings. In addition, each takeoff rating can be further reduced by use of an assumed temperature (flex for Airbus guys).
Typically when programming takeoff thrust, our FMS will automatically select a logical climb thrust (for example, a TO2 takeoff with an assumed temperature would probably result in the FMS "suggesting" a CLB2 rating). This climb rating can be modified by the pilot.
If, say, I plan a D-TO2 53˚ (takeoff thrust derate two with 53 degrees of assumed temperature) and then on the FMS I pre-select CLB1 or even full CLB (because of terrain or other gradient requirements for example), then when climb thrust is selected the thrust will actually increase.
As I recall this was not possible on the A321 instead, where if projected CLB thrust was greater than flex takeoff thrust the takeoff thrust was "bumped up" automatically by the FMGCs to equal climb thrust: in that case when climb thrust was set the thrust didn't actually change.
As far as derated climb thrusts being increased as the climb progresses, again it depends: on some of our PW-engined aircraft the climb derate "washes out" by 15,000', and you have full climb thrust fairly soon. On our other PWs and on our GE engines the derate washes out more progressively between 10,000' and 30,000' or even 35,000'. On our RRs it is different yet again, and on CLB2 might not even wash out entirely.