Some graphs, facts and analysis:
Basically, it's a cluster F, but it didn't just start under Biden, shifts were already happening during Trump, some of the policies he put in place have made it worse, and not enough is happening fast enough to get under control quickly. May not even be realistically possible in the short term.
https://www.wola.org/analysis/putting-border-crisis-narrative-into-context-2021/
There are real politically-independent drivers pushing migrants to the US. Economics, natural disasters (eg recent severe hurricanes, covid) and home country crime and politics.
Then add mean spirited Trump policies, that were no more effective than squeezing a water balloon. Close up amnesty programs, force people to stay in Mexico, illegal border crossing go up as migrants try other ways to get in.
Unlike the past, recent migrants are not Mexicans in origin, but they are piling up in Mexico. Venezuelans are currently the highest number of non-Mexican migrants apprehended.
The chances of being returned to their home country depends on diplomacy and cost. Basically, bigger the PITA, more likely they don't get sent all the way back (and just get tossed into Mexico.)
Basically, we'll need to continue to work diplomatic fronts with Mexican partners to improve security there, meanwhile getting home countries to take their citizens back, all the while trying to unfuck the amnesty/legal migration pathway until the situation can be stabilized.
Biden working on his relationship with the border patrol agency also needs to happen as well. Lots of eye poking back and forth since the horse incident.