Any large unexpected influx of people causes havoc on a city. It strains hospitals, social services, schools, etc. It's becoming a problem in places far from the border, but border towns at commonly used crossings get hit first, harder, and repeatedly. People overstaying their visa don't swarm into towns in massive numbers in short windows of time.

There is no shortage of news stories about the issues.

Examples:

https://www.tpr.org/border-immigration/2023-05-01/texas-bord...

https://www.texastribune.org/2019/07/03/texas-cities-confron...

https://www.valleycentral.com/news/local-news/small-south-te...

https://doctorsoftheworld.org/blog/humanitarian-crisis-the-m...

https://www.texastribune.org/2023/09/21/texas-migrants-borde...

https://www.npr.org/2024/02/19/1232383832/schools-face-big-c...

https://nypost.com/2023/09/20/eagle-pass-texas-declares-stat...

https://www.npr.org/2023/04/02/1167645456/border-towns-are-f...