The extent to which other former Soviet countries interact with (or join) the Customs Union has been, and will continue to be, driven less by the incentives of joining the bloc than by the relationship these countries have with Russia and their position in their wider regions. Each potential member of the union thus presents a special case for Moscow to deal with if it wishes to expand the Customs Union and Eurasian Union to include more former Soviet states.
Ukraine — the most important former Soviet territory to Russia — is an interesting example of Russia's use of the Customs Union. Moscow has spent a great deal of effort trying to persuade Kiev to join the Customs Union. However, Ukraine is difficult for Russia to deal with because of its internal divisions. Ukraine under President Viktor Yanukovich has been careful to retain a certain degree of distance from Russia, instead pursuing a "dual vector" foreign policy in which it cooperates with Russia and the West (particularly the European Union). In this regard, the Ukrainian government has been noncommittal toward the Customs Union. Ukraine initially said it was not interested in joining and preferred a 3+1 format in which it would form a free trade agreement with the union short of membership. More recently, Yanukovich said the country wants to see the "achievements" of the Customs Union before making any major decisions on Ukraine's role in it.
But the nature and timing of that decision may not be up to Ukraine. Because of domestic political issues, Ukraine has become alienated from the European Union. Thus, the bloc no longer serves Kiev as a potent counterbalance to Russia. Financial issues due to rising energy costs from Russia have put Ukraine into an economic bind that, combined with an alienation from the European Union and International Monetary Fund as financial resources, leaves the country with few options. It is thus no coincidence that Russia has offered to lower natural gas prices for Ukraine on certain conditions, such as Kiev ceding its natural gas transit operator Naftogaz to Gazprom or joining the Customs Union. Either would undermine Ukraine's sovereignty (which is in a sense Russia's goal).
Ukraine's ability to resist Russia's demands will depend on the country's domestic political and economic situations, neither of which appears to have a positive trajectory for Yanukovich. Ultimately, acquiring Naftogaz matters more to Russia than having Ukraine in the Customs Union; the latter would simply be a way to facilitate the former, among other goals.
Kyrgyzstan has applied for Customs Union membership, and Tajikistan would be the next logical member after that (and Tajikistan has said it will only apply after Kyrgzstan is in). However, Dushanbe's membership is far from assured, because the Tajik government is not as close to Moscow as the Russia-backed (and recently imposed) regime in Bishkek, and Russia and Tajikistan have had frequent spats on the economic and diplomatic levels. Although Tajikistan is strategically oriented toward Russia in the security sphere (Russia has three military bases and 7,000 troops in the country), there have even been disputes in this area. Dushanbe resisted leasing the Ayni Air Base to Russia and has not allowed Russian troops to return to patrol the Tajik-Afghan border. Therefore, while Tajikistan's accession to the Customs Union is certainly possible, it may come only after some concessions that Tajik President Emomali Rakhmon will not be eager to give. The more important issue to Russia is Tajikistan's security orientation.
Even less likely to join the Customs Union are Uzbekistan and Turkmenistan. Independently minded Uzbekistan has long resisted Russia's integration blocs, calling them "politically motivated." Indeed, Tashkent recently left the Collective Security Treaty Organization and is rumored to be looking for closer security cooperation with the United States or possibly China. Uzbekistan's membership in the Customs Union would therefore be unlikely and would come only as a result of severe pressure from Russia. Moscow is more interested in limiting Uzbekistan's security relationship with the United States and China. Turkmenistan is averse to any sort of alliance (it is not in the Collective Security Treaty Organization and is only an unofficial associate member of the Commonwealth of Independent States). Ashgabat and Moscow have a quiet understanding that Turkmenistan's membership in the Customs Union is not necessary as long as its cooperation with other outside powers is rare.
Armenia is one of the most loyal countries to Russia, hosting a Russian military base and depending economically on Moscow. But its reaction to the Customs Union has been interesting in that it has been a dual response. President Serzh Sarkisian has said Armenia could join the union and has asked about paths to membership, while Prime Minister Tigran Sarkisian has said the country should not join because it does not share any borders with Russia (though he has advocated for a special form of cooperation with the Customs Union, along the same lines as Ukraine's proposal). Armenia seems to have no political motivation or sense of urgency to join the Customs Union, and Russia has not seemed to mind. This could reflect a certain amount of comfort Russia has with its degree of influence in Armenia — even more so than in bloc members Belarus and Kazakhstan (which have not always been the most loyal or friendliest to Moscow).
Georgia, firmly oriented toward the West, will surely continue to resist membership in the Customs Union and any other Moscow-dominated blocs. But Russia is more interested in maintaining its military presence in the breakaway republics of Abkhazia and South Ossetia than in getting Georgia to join the Customs Union.
Azerbaijan is a more interesting case, because it has a more complex and nuanced relationship with Russia than its Caucasus neighbors. Given Russia's support for its archrival Armenia, Azerbaijan has stayed out of Moscow-dominated security blocs (but it is a member of the Commonwealth of Independent States).
In terms of the Customs Union, Azerbaijan has recently announced that it would not join the union but would instead implement its own customs code and seek to integrate customs with Turkic countries such as Turkey and Kazakhstan. While the plan is fraught with complications, it does exemplify Azerbaijan's strategy to diversify its options and not get too close to Russia. Therefore, barring a dramatic realignment of forces in the Caucasus, Azerbaijan's membership in the Customs Union is unlikely. But Russia is more concerned with making sure Azerbaijan does not serve as the source of energy diversification to Europe than with Baku's membership in the Customs Union.
The Baltics and Moldova
Other former Soviet countries like the Baltics are already in the European Union and are certainly not interested in joining the Customs Union. They will resist Russia's advances no matter what happens in the eurozone. Following the collapse of the Soviet Union, their economies were largely directed away from Russia (with the exception of the Russian energy supplies they are trying to wean themselves from), and they are closer oriented to Nordic countries — financially and economically — than to Russia. Therefore Russia's strategy here has been more to disrupt further integration rather than try to orient the Baltics toward Moscow, an unlikely prospect.
Moldova under its current government is oriented toward the West, though the country retains a substantial Russian presence and orientation. At best, the country is split, meaning that any significant integration of Moldova proper (Transdniestria is essentially a satellite state of Russia) into the Customs Union is unlikely in the near to midterm. Therefore, keeping Moldova divided is Russia's top priority, and advocating the Customs Union to the pro-Russian elements in the country would be a useful tactic in doing so.
The Eurasian Union's Future
The Eurasian Union is just one part of Russia's overarching strategy as it tries to build what it considers sufficient leverage in its near abroad. Developments in relations between Russia and its former Soviet states in the next two and a half years will determine the degree of Moscow's success as it uses the institution to spread its influence throughout its former Soviet states.