The second-wave of the Polish Muscovite War was declared by Polish king Sigismund III in response to Tsar Vasili declaring a military alliance with Sweden. The war had two goals. The primary goal was to weaken Sweden's ally and to gain territorial concessions. A secondary goal was to also Catholicize Russia, a goal which was supported by the Pope.
What made Russia even more inviting was that Russia was in absolute chaos as a result of the Troubles, so the Russians were fighting over a new Tsar.
Russian boyars had offered Sigismund support by supporting his 15-year old son, Prince Wiadyslaw, for the position of Tsar, hoping that the Polish–Lithuanian Commonwealth would end the despotic rule of their current tsars, by having Wiadyslaw as the Tsar of Russia.
The Polish Army won several early victories and had even reached Moscow. However, there was a fine print to the boyars' support. They had also pledged to support Wiadsyslaw for the throne if he converted to Orthodoxy which made Sigismund furious.
Eventually, Wiadyslaw was elected Tsar, but Sigismund stabbed his supporters in the back by then declaring himself Tsar. Knowing that Sigismund wanted to convert the Russian masses from Orthodox to Catholic, the boyars soon changed sides and the Polish presence in the country was soon met with hostility.
A 1611 uprising in Moscow would soon end any Russian support for Polish-Lithuanian Intervention in the Troubles.
But let's imagine that the Russian boyars never forced the stipulation of Wiadyslaw becoming Orthodox and he was allowed to become a Catholic Tsar, but a puppet to his father, Sigismund III.
So, how would Sigismund III, through his son have ruled Russia? Would he have violently supressed all forms of Russian culture and tried to force Polish culture onto the populace or would the conversion of the Russian populace be a slow and gradual process?