How, exactly, does the 'input signal' wave(s) modify the carrier-wave of a radio signal?
The process of getting an input signal to modify the carrier-wave of a radio signal is known as modulation. The input signal is called the baseband signal or modulating signal. The combined signal is called the modulated signal.
Wikipedia has an introductory article on modulation which briefly reviews a wide range of modulation types. You might want to follow up with SteveSh's excellent suggestion of looking at AM as a straightforward modulation scheme.

Is it entirely due to constructive and destructive interference?

No. Modulation doesn't work by just adding the instantaneous values of the modulating and carrier signals together. The mathematical description varies between modulation types.  AM is the simplest. (If you don't get on with that Wikipedia description, as SteveSh points out, there are plenty of others.)

Also, are the carrier-wave and modifying wave created and broadcast on
all the same equipment?

It varies but the number of equipment boxes doesn't affect the principles of modulation. You could do everything in a single box, like a mobile phone, or split the transmitter functions across multiple boxes.
The modern approach, used across the full range of modulation types, is to perform the modulation at low power after which the modulated signal is amplified in the transmitter and possibly a separate power amplifier. (As you've added a ham-radio tag the power amplifier is often known as a linear in amateur radio circles.)
The modulation might be done in a separate box known as an exciter, which will send a low power modulated signal to a separate transmitter to be amplified then sent to the antenna. Or the exciter and RF amplification functions can be combined in a single box, also known as a transmitter.
An older technique, used in older ham AM (and other modest power) transmitters is high level modulation where a baseband audio signal is amplified to a high power level in a separate audio amplifier chain and is used to modulate the RF carrier at high power in the final stage of the transmitter.

Broadcast using the same antenna?

Once the baseband signal and the RF carrier have been combined in the modulation process only a single modulated signal is sent to the antenna. There are no longer two separate signals to send to the antenna.