A transmission line with continuously varying impedance, how would reflection occur in this case?
What you're asking about is called a transmission line taper.
In general, there's no analytical solution to describe the reflections. The link in Chris L's answer (if you follow through to Klopfenstein's paper) gives some examples of specific taper shapes where something close to an analytical answer has been found.
The basic way to study it is to imagine breaking up the continuous taper into several segments, each with a slightly different Z0 value. You calculate the reflections at each discontinuity and how they add up to give the overall reflection and transmission characteristics.
Then you split up the taper into finer and finer steps (with smaller and smaller discontinuities in Z0) until you have a good enough approximation to the continuous taper. You could try to calculate the results by hand but it's much easier to just get a computer program to do do it. Luckily, this kind of program is pretty easy to find --- it's called a finite element simulation program.
Continuous varying impedances are used all the time for impedance matching. If you have a very capacitive part of a trace (for example, where a large component pad might be), you can have a relatively inductive transition before or after it to "balance" it out.
What will end up happening is that the reflections will "stack up" but, instead of being at one point (a VSWR peak), it will be moderately spread out. You can still imagine it discretely, but in small steps.
And also remember, if you have a small reflection point, any backward reflection after THAT will be reflected slightly FORWARD, and so on.
Anyway, the good gents at http://www.microwaves101.com/encyclopedia/klopfenstein.cfm always have a nice, in depth explanation.
edit: I didn't completely answer your question. "How it would look" is dependent a bit on how you are describing it. In the frequency domain, what you'll probably get is a VSWR that is "de-Q'd". You'll go from a nice sharp peak at midband to a more gradual, broader band response.
In the time domain....well, I don't work with the time domain as much but I would imagine you would have a lower amplitude, longer pulsewidth "ringing" or reflection.