MOSCOW (AP) _ Tensions mounted in several southern republics Tuesday as lawmakers in Moscow debated scrapping the vestiges of a strong central power that had long suppressed the ancient rivalries of the Soviet periphery.

Many republics and minority groups gained momentum from last month's collapse of a hard-line coup against President Mikhail Gorbachev, reasserting aspirations for independence, greater rights and reforms.

In the latest of the fast-paced changes sweeping the nation, Gorbachev and 10 other republic leaders this week agreed on a new political system that would strongly limit the power of the central government.

In the southern republic of Georgia, thousands of demonstrators opposed to the republic's Georgian nationalist leadership took to the streets Monday and a clash left as many as 30 people injured, reports said. Another opposition rally was scheduled for Tuesday night.

Centuries-old tensions between mostly Christian Armenians and Muslim Azerbaijanis simmered in the disputed enclave of Nagorno-Karabakh. And political movements in Azerbaijan and Georgia moved to oust republican leaders they claim sided with the coup plotters.

In Moldavia, which is dominated by ethnic Romanians, minority Slavs blocked the railway in an attempt to gain the release of a jailed leader who had headed their secession movement.

The government of Azerbaijan on Tuesday condemned Nagorno-Karabakh and the neighboring Shaumyan district, where the people are predominantly Armenian, for declaring themselves an independent republic. Azerbaijani deputy Yashar Abbasov said the move was meant ''to torpedo the first signs of dialogue between the Azerbaijani and Armenian communities.''

Vladimir Akopian, secretary of Shaumyan's executive council, said the Armenians want to completely separate from Azerbaijan and put themselves under the protection of the Russian republic.

Both Azerbaijan and Armenia declared independence last week. But Azerbaijan claimed the right to negotiate directly with Nagorno-Karabakh on its future.

The tensions between Armenia and Azerbaijan, neighboring republics in the Caucasus between the Black and Caspian seas, erupted in violence in 1988. Hundreds have died in sporadic fighting, and sporadic shelling continued this week in its Shaumyan region, Akopian said.

Meanwhile, the Azerbaijan Popular Front called a strike Tuesday across the republic to back its demand that presidential elections scheduled for Sunday be called off, the Russian republic's information agency reported.

The front later reported that workers at 62 enterprises in the republic's capital of Baku had struck, but support had not spread further.

Azerbaijani President Ayaz Mutalibov, a longtime Communist, has been accused of supporting the coup. The front is demanding Mutalibov's resignation, a ban on the Communist Party and multi-party elections.

In Georgia, the opposition National Democratic Party said police opened fire Monday on several thousand demonstrators in the capital, Tbilisi. The demonstrators were demanding the resignation of President Zviad Gamsakhurdia, who the opposition claims supported the attempted coup.

Gamsakhurdia said early Tuesday that the protesters started the violence by firing on police. He accused the opposition of aiding a Kremlin attempt to thwart Georgian independence. Reports varied on the number of injuriies ranging from eight to 30.

In Moldavia, tensions continued over the proclamation of a ''Trans- Dniester'' republic by minority Russians and Ukranians living in the Tiraspol region.

Last Thursday, Moldavian police went into the neighboring Ukraine to arrest Igor Smirnov, chairman of the Tiraspol City Council and leader of the self- proclaimed republic's legislature. Pickets demanding his release have blocked rail traffic.

Moldavian officials said Smirnov would be charged with ''inciting confrontation between ethnic groups'' and faced up to 15 years in prison.

Viktor Pushkash, deputy chairman of the Moldavian Parliament, was quoted by Tass as saying the blockade already had caused ''considerable economic damage to the republic as a whole'' and could get worse if the ''Dniester rebels'' continue their activities.