My friend lost her HKSAR passport, how can she return to HK from Chile, via the US or transiting elsewhere?

According to flightconnections.com, the airports that have non-stop service to both Santiago and Hong Kong (and so could be used for a connecting city) are: Sydney (SYD), Auckland (AKL), Madrid (MAD), Paris (CDG), Rome (FCO), Toronto (YYZ), London (LHR), Dallas/Fort Worth (DFW), New York (JFK), and Los Angeles (LAX).

For these countries, according to Wikipedia, a Chinese citizen of Hong Kong with a normal passport requires the following documents to enter these countries:

• France, Italy, or Spain (i.e., Schengen), the UK, or New Zealand: No visa necessary
• Canada & Australia: an electronic travel authorization (eTA)
• United States: full visa required

Note that the restrictions may be different for temporary passports, or for transit as opposed to entry; so I would not purchase any new ticket until you know precisely what kind of travel document you will have and you've triple-checked its acceptability with the airlines.

It is also possible to connect via two stops in Brazil (either São Paulo or Rio) and thence to either Johannesburg, Doha, or Dubai; if none of the above options work out, these might be worth investigating.

If your friend is issued a conventional Chinese passport as a temporary replacement, rather than a HKSAR passport, then according to the excellent answer here, she should still be able to transit in the Schengen area so long as she does not need to leave the international area of the airport. The UK, USA, Australia, New Zealand, and Canada will all require transit visas.

She cannot transfer at DFW, or any US airport, without a visa. There is no such thing as "sterile transit" in the US; in other words, anyone landing in the US is required to clear US immigration, even if they are immediately proceeding to another flight out of the US.

Lost passport aside, HKSAR passport holders are not eligible for the Visa Waiver Program, so your friend would need some type of US visa (most probably a B1/B2, or a C visa).

European airports are generally better for sterile transit than US airports, and there are numerous flights from Santiago to Hong Kong transiting Europe. But you/she would need to look closely at the transit requirements, depending on exactly what kind of replacement document she gets issued (as HKSAR passports generally have much better visa-free privileges than PRC passports, but it's not clear in this situation what she'll get.)

She can also fly from SCL airport in Chile to Hong Kong on Qatar Airways passing through Sao Paulo and Doha with no transit visa required for a Chinese passport holder.