Very little is known of how the first fighting between the balrog and the dwarves of Moria went. In fact, it appears that the only information may be what is given in Appendix B ("The Tale of Years") of The Lord of the Rings:
1980 ... A balrog appears in Moria and slays Durin VI.
1981 Náin I slain. The Dwarves flee from Moria....
Clearly, there was not one single battle in which Durin's Bane earned his name by killing the king and driving the other dwarves away. The dwarves remained, presumably struggling with the balrog, for at least a matter of months, until they lost another king. Only after that did Thráin the Old lead them away from Khazad-dûm. However, what went on during this period of conflict was never revealed, and we do not know what tactics were used by either side in the conflict.
We do get one further glimpse of the balrog's power of the dwarves, almost nine hundred years later, however. According to Appendix A, at the Battle of Azanulbizar:
Up the steps after him leaped a Dwarf with a red axe. It was Dáin Ironfoot, Náin’s son. Right before the doors he caught Azog, and there he slew him, and hewed off his head. That was held a great feat, for Dáin was then only a stripling in the reckoning of the Dwarves. But long life and many battles lay before him, until old but unbowed he fell at last in the War of the Ring. Yet hardy and full of wrath as he was, it is said that when he came down from the Gate he looked grey in the face, as one who has felt great fear.
Then Thráin turned to Dáin, and said: ‘But surely my own kin will not desert me?’ ‘No,’ said Dáin. ‘You are the father of our Folk, and we have bled for you, and will again. But we will not enter Khazad-dûm. You will not enter Khazad-dûm. Only I have looked through the shadow of the Gate. Beyond the shadow it waits for you still: Durin’s Bane. The world must change and some other power than ours must come before Durin’s Folk walk again in Moria.’
So even Dáin Ironfoot, one of the greatest dwarf warriors and kings of the Third Age, was so terrified by the mere presence of the balrog inside Moria that he could not enter, and it seems that he knew that no other dwarves would be able to enter either—at least, not at that time. When the Company meet Durin's Bane, the witness that
a power and terror seemed to be in it and to go before it,
and it seems that was an exceedingly powerful weapon in itself.