What you're referencing is a "Cavalry-Mechanized" Group, and the co-operation between horsemen and tanks was not as close-knit as you may think. They were not akin to the "Horse-Mechanized" regiments^1 of pre-war America.

Rather, a Cavalry-Mechanized Group was a miniature army-sized formation generally consisting of a Cavalry Corps, or a series of Cavalry Divisions (small units of less than 10, 000 men in reality) amalgamated into a Corps-like formation. They would then be coupled with a traditional Mechanized Corps, made up of Tank Brigades and perhaps a Motor Rifles division to form the Cavalry-Mechanized Group. These formations were generally second and third echelon formations in Soviet Doctrine, and would not be committed to battle until Operational Complexity^2 had been achieved. In short, these were small forces meant to exploit breaches. Thus a premium was placed both on self-sufficiency and speed. Cavalry, to a relative degree, had both; a peculiarity of fighting East of the River Bug with space and fodder aplenty.

Cavalry was a mainstay for many reasons, largely the fact that they were less of a logistical strain on the Steppes, but also because the Russians lacked a tracked infantry carrier outside of lend-lease and could not put forward the necessary amount of men by simply riding on tanks (though this was a common practice as well). Tertiary factors include the fact that the Soviet Union had many 'natural' riders whom did not need to be trained on the basics of horsemanship and the simple reasoning that a Horse can go where a Tank can't in marshy, wet, or flooded Steppes.^3

As aforementioned, the Cavalry units and Tank Brigades rarely worked in the closest of combined-arms support, that is at the tactical level. However, Cavalry often had organic tank units in their division or Corps structure. ^4 So, no, they wouldn't necessarily ride with the Tanks, and Red Army cavalry generally fought dismounted in this period. They could, however, keep up with the Tanks in a far greater manner relative to a Rifle Corps. They were an excellent 'tether' between the forward moving Mechanized Corps and the mopping up Rifle formations.

---

^1: The idea of "Horse-Mechanized" units, the brain-child of the stubborn John K. Herr, Chief of US Cavalry, was a last gasp attempt by this old horseman to keep his beloved cavalry in saddle without appearing to be reactionary to the idea of mechanizing. The fathers of the modern US Cavalry/Armor, naturally, vehemently protested this organization and rightly declared it as faulty for a nation with the industrial capacity such as America. Tactical considerations of their fault soon followed. Chaffee, after seeing a rather disastrous exercise of one formation, concluded: "any consideration of brigading or divisioning horse and mechanized elements is faulty organization. You can just as well put a yoke around the tortoise and the hare and expect them to race as a team." Folksy? Perhaps. Succinct, never the less. (Hoffmann, Through Mobility we Conquer, page 240)

^2: Suggested further reading

^3: David Glantz in his Colossus Reborn posits that the continued dependence on Cavalry may also have had a romantic element attached to it. His thinking-aloud refers, of course, to Stalin's old command of a Cavalry unit and his belief/affection in their continued relevance. The fact that the RKKA still had 800, 000 cavalrymen on call is probably a greater factor. One cannot discount the whims of a dictator in an absolutist state, however.

^4: On page 262 of Colossus Reborn we are provided with the Table of Organization and Equipment of a Red Army Cavalry corps, the numbers for a mid-war (Summer, 1943) formation confirms the presence of organic tank support. Paper strength of 21, 000 men, including in its materiel 69 T-34s and 48 T-70 (light) tanks. Minute compared to a Mechanized Corps, but present in sufficient numbers to provide tactical-level fire support to the Cavalrymen.