After seizing power and then needing to defend it, the Bolsheviks faced a dilemma. They had always openly espoused a citizen-led army, and often extolled the virtues of irregular operations compared to traditional armed forces. But once fighting started in earnest in the civil war it became clear that partisans were both ineffective militarily and were a major challenge to them politically.
The result was a two-faced strategy. Partisans behind enemy lines were supported politically, and occasionally materially, if they were not opposed to Soviet power. Once the Red Army reached them, however, any partisan unit was quickly absorbed into the regular army – and its leaders sent elsewhere to break the personal connections.
Bolshevik propaganda after the war tended to gloss over this process. To show that the Communists had won the war due to their popularity with the people it tended to describe any partisan operations behind White lines as being Red, when most were as firmly anti-Bolshevik as they were anti-White – the so-called "Greens" (or if anarchist, "Blacks"). But in the Soviet nomenclature of the time only pro-Soviets get called partisans, and those that refused to join the Red Army once it reached them were instead "bandits" or "gangsters".
To show how much partisan activity was taking place behind Soviet lines, we can look at the order of Western Front Commander Egorov for 23 March 1920
... 7) In view of the acute situation of the internal front, given that it will intensify even more with the beginning of summer, when gangs of bandits will have the opportunity to widely develop their actions, hiding in the forests, strengthening the rear of the front is a matter of great national importance no less than the position of the external front. At least four divisions are required for this. ...
In the Ukraine the Red-leaning partisans were absorbed in the initial drive across the Ukraine by the Red Army. When the Soviets fell back those that remained were largely nationalist (mostly UNR) or were attached to Makhno, Zelenyi or the like, because known Communist activists had a short lifespan in White held territory. Denikin did not face large organised Soviet resistance in his rear – although there was more than enough Greens and Blacks, of course.
Between the Ukraine and the Urals any Red partisans were also quickly absorbed into regular units early in the war. In particular the Soviets were desperate for cavalry, so any "Red" Cossacks were used as the basis for that arm.
Some of these units continued to have "partisan" in their names long after they were regular army, which does confuse matters.
It was only in Siberia where resistance groups could not link up to the centre and so operated well into the civil war outside the Red Army. Later Soviet literature greatly enlarged the significance of the partisans such as Kravchenko and Shchetinkin there, as if they were serious military threats in the manner of Makhno. However, most Red partisans were entirely negative, in that they largely denied Whites access to their local area by ambushes and threats to their supply.
Being desperately short of weapons and ammunition, with little military training and no firm command structure, the typical Siberian Red partisan groups were incapable of much by way of active operations and could not stand in the field against any significant opponent. They did tie men down on the rail-line, but since it was largely Czecho-Slovaks, Japanese and Americans, who could not be moved to the front anyway, the war effect was minor.
As in WWII France, the massive growth in partisans occurred only once the war was clearly nearly over.
The puppet "Far Eastern Republic" was established in April 1920 to allow the Soviets to deal at arms' length with the Japanese and remaining Whites. Its armed forces were, in effect, the Red Army. In order to allow itself to break some of their agreements, these men were sometimes required to pretend to be partisans.
In the very early war any available unit regardless of quality was thrown at the front, but after that it was rare to see partisans fighting alongside regular Red units. Sometimes "partisans" were listed in regular formations. I believe these are irregular units sent behind enemy lines, so operating independently, or simply former partisans whose unit had not yet been renamed.
Partisans were almost always very poorly armed, sometimes not having rifles at all. Some were cavalry.
The one major exception was during Nikifor Grigoriev's campaigns in the Ukraine in early 1919, when he managed to take control of some regular Red Army alongside his units originally formed as partisans. Because he was able to arm and supply them, they became capable of operating outside their local area. Of course, they weren't actually fighting for the Soviets.
Pavel Dybenko may have done the same in the Crimea, when he set up his bizarre "Crimean Soviet Army".
Makhno did ally himself a couple of times with the Soviets, but he always kept his forces completely separate, and it would be anachronistic to mix his units with Red Army ones.
Siberian Partisans in the Civil War by David Footman, St Anthony's Papers Number 1, Soviet Affairs Number One, 1956.
One of the few non-Soviet discussions about partisans. He is quite dismissive of their prowess.