I've operated mostly R985's, R1340's, R2600's, R3350's, and R4360's. Smoking on start is due to oil, primarily in the lower cylinders, but also due to excessively rich mixtures and a lot of unburned fuel initially passing through the engine.
Unlike a turbine engine in which one or several burner cans can typically be lit at the same time and easily, the radial engine may have 9 to 28 cylinders to individually light off.
During the start process as prime is applied, it's dumped into the supercharger as raw fuel, slung around the supercharger, and dumped into cylinders. Even in a small piston engine, mixture isn't consistent from cylinder to cylinder, and the mixture needs to be within an easily ignitable range for the cylinder to fire. Too rich or too lean, and it wont' fire. During start when airflow is low through the carburetor, establishing a consistent mixture is compounded a lot more for each of the cylinders on the engine.
The spark plug in each cylinder needs to be able to fire, and plugs that are fouled with lead or oil tend to fire weakly or not at all. The end result is that some cylinders will fire while others won't.
You may have noted that some radials tend to catch fire on start more than others. This is partially due to starting techniques, and partially due to the construction of the engine itself. Particularly in the case of cold weather starts, large amounts of prime are sometimes used, and improper starting techniques can result in a lot of gasoline and fires. On some motors, we would have five gallon buckets under the engine during the start to catch fuel pouring out of the supercharger drain, as the technique involved flooding the engine then starting it off the mixture.
Unlike "modern" horizontally opposed recip engines, the radial offers a fairly poor induction distribution system with respect to establishing a consistent mixture at each cylinder. This is more pronounced at start. The first few blades (counted as blades passing a given point, by whomever is turning the starter on a particular airplane) are done with ignition off to clear the engine, ensure no hydraulic or liquid lock (particularly in the bottom cylinders), and to begin drawing fuel into cylinders (eventually); the initial part of the start looks slow to an observer because the engine is being turned through a number of revolutions before the start actually begins. In some cases, this is done by hand to protect the engine, but in most cases it's done with the starter to protect the engine (because the starter clutch will slip before bending a connecting rod in the event of a hydraulic lock). The particulars depend on the engine and the operator practice.
Once fuel and ignition is introduced, each cylinder must be individually awakened. This isn't necessarily a fast process. Whereas there's a lot of mass for the starter to turn over, it doesn't happen quickly, the mags aren't turned rapidly, and mag points don't open rapidly at first, and spark isn't necessarily as hot. Plug fouling and inconsistent mixture mean that each cylinder will "wake up" at a different moment, and enough cylinders need to be firing (and firing consistently) enough to keep the engine turning. The result is that the engine tends to lope and cough on start, belching smoke, often banging and after firing (and in some cases, undesirably, backfiring (the difference between burning through the exhaust or through the induction; a backfire can separate induction tubes and cause an engine fire, whereas an after fire does not).
Starting some radial engines is part science, part art. The start can vary, depending on how much oil is in the cylinders. Some engines can be dry consistently, and then one start be locked or wet ("wet," with a radial, can be expressed in terms of both excess fuel, and oil) on the next. Some engines can sit for months and never liquid lock, while others can sit for five minutes and have hydraulic lock problems, or oil pooling in the lower cylinders. For some radial engines, especially in cold weather, starts can be more art than science.