What are the main challenges involved in transferring money across borders?

There are many reasons for this. If you are asking specifically with respect to a business (of international / cross-border money transfer) then consider the following factors (in no particular order):

• Typically for each corridor (a corridor is the transfer between two countries, examples of corridors are US-Mexico, UK-India, Saudi Arabia-Philippines, etc.) you have to have a complete arrangement for funds to be transferred from the originating source to the beneficiary destination. This means, until and unless you don't have a "route" (financially speaking), making a transfer is then not easy.
• Licensing: Non-Banking institutions sometime have a very tough time getting licensed as Licensed Tie-ups in countries they would like to work with. In very many countries, just because you are licensed in say UK, you cannot just unilaterally send funds to India. There is, in place an entire procedure to legally obtain permission to enter another country for purposes of money transfer.
• AML/KYC limitations and/or short-comings. Many a times, the counter party or the financial regulator will not accept the AML/KYC procedures you have in play and will request that you to improve upon them as well as the reporting parameters, so as to reapply and get certified.
• Inadequate capital, in some corridors you are required to pre-fund your accounts rather than after-realization of funds method.
• Tighter controls: In some markets, especially competitive markets, players are restricted by (a) either the financial regulator (b) or by the existing players themselves (or a combination of both), so as not to cause over-saturation of players in the market and reduce the existing margins, etc.
• Access to settlement banks: Sometimes all the existing settlement banks are already engaged with various tie-ups, and might not be able to take additional cross-border termination clients.
• Inadequate knowledge about a specific country. If for example a company in the UK wants to engage with a on-ground settlement partner in say Zimbabwe, they would need to understand the payments eco-system on ground to be able to correctly have access to that market. In most cases, access to the right people advising you is usually not so easily available.