Before considering how broadening arises, it is worth mentioning that when multiple sources of broadening occur together, the each broadening contribution acts on each other in a process called convolution, which is reproduced mathematically by a convolution integral, a fairly slow computation. Use of a Fast Fourier Transform speeds this computation, because taking two functions, Fourier transforming them, multiplying the transforms together and then taking the inverse transform of the product is equivalent to performing a convolution integral, but even that is a fairly involved computation.
I like to think about broadening as an analogy to optical aberrations. Broadening is in effect a blurring of the peak position that comes from multiple compromises in the design of a diffraction instrument, as well as possible imperfections in the instrument, as well as defects in the sample. For those of you who, like me, must wear glasses to see well, we know that dirt on the lenses degrades our ability to see clearly. Bright sunlight, which allows the iris of our eye to become more like a pinhole camera, decreases the impact of this schmutz. As an analogy for how broadening sources combine, consider what you see looking out a glass window. If the window glass is not uniform in thickness, then your view of the outside will be distorted where some aspects of the view will be enlarged, and others decreased in size since the window will act like a lens. If in addition, it is raining and the window is covered with rain, then the view will also be blurred, so your view of the outside will be both blurred and distorted. However, if the rain is heavy enough then the view will be so blurry that the distortions become hard to distinguish. A similar effect happens with broadening, where if we have two types of broadening that have relatively similar impact, then the two effects largely (but not exactly) add. If one broadening term is causing significant change to the signal and the other causes much less impact, then the convolution product will be only slightly changed from the more significant broadening source.