The problem is with the UILabel content mode. You see a similar problem here. Because you are shortening the label where the text is drawn, the label does a drawing pass, but it uses the final frame of the animation block for the update. The easiest way create a smooth animation is to use a greaterThanOrEqualTo leading constraint and set the textAlignment to .natural. 
When I do that I get the result on top as apposed to setting textAlignment .right on bottom.

Source code for demo:
import UIKit
import PlaygroundSupport
final class ViewController: UIViewController {
    var constraint: NSLayoutConstraint?
    var otherConstraint: NSLayoutConstraint?
    lazy private var sideView: UIView = {
        let view = UIView()
        view.translatesAutoresizingMaskIntoConstraints = false
        self.view.addSubview(view)
        view.topAnchor.constraint(equalTo: self.view.topAnchor).isActive = true
        self.constraint = view.leadingAnchor.constraint(equalTo: self.view.trailingAnchor)
        self.constraint?.isActive = true
        view.widthAnchor.constraint(equalToConstant: 100).isActive = true
        view.heightAnchor.constraint(equalToConstant: 44).isActive = true
        view.backgroundColor = .yellow
        return view
    }()
    lazy private var secondSideView: UIView = {
        let view = UIView()
        view.translatesAutoresizingMaskIntoConstraints = false
        self.view.addSubview(view)
        view.topAnchor.constraint(equalTo: self.view.topAnchor, constant: 44.0).isActive = true
        self.otherConstraint = view.leadingAnchor.constraint(equalTo: self.view.trailingAnchor)
        self.otherConstraint?.isActive = true
        view.widthAnchor.constraint(equalToConstant: 100).isActive = true
        view.heightAnchor.constraint(equalToConstant: 44).isActive = true
        view.backgroundColor = .yellow
        return view
    }()
    lazy private var label: UILabel = {
        let label = UILabel()
        label.text = "Label"
        label.translatesAutoresizingMaskIntoConstraints = false
        self.view.addSubview(label)
        label.topAnchor.constraint(equalTo: self.view.topAnchor).isActive = true
        label.leadingAnchor.constraint(greaterThanOrEqualTo: self.view.leadingAnchor, constant: 8.0).isActive = true
        label.trailingAnchor.constraint(equalTo: self.sideView.leadingAnchor).isActive = true
        return label
    }()
    lazy private var secondLabel: UILabel = {
        let label = UILabel()
        label.textAlignment = .right
        label.text = "Label"
        label.translatesAutoresizingMaskIntoConstraints = false
        self.view.addSubview(label)
        label.topAnchor.constraint(equalTo: self.view.topAnchor, constant: 44.0).isActive = true
        label.leadingAnchor.constraint(equalTo: self.view.leadingAnchor, constant: 8.0).isActive = true
        label.trailingAnchor.constraint(equalTo: self.sideView.leadingAnchor).isActive = true
        return label
    }()
    override func viewDidLoad() {
        super.viewDidLoad()
        view.backgroundColor = .white
        self.secondSideView.isHidden = false
        self.secondLabel.isHidden = false
        self.sideView.isHidden = false
        self.label.isHidden = false
    }
    override func viewDidAppear(_ animated: Bool) {
        super.viewDidAppear(animated)
        self.perform(#selector(self.showSideView), with: nil, afterDelay: 5.0)
    }
    @objc private func showSideView() {
        self.view.layoutIfNeeded()
        UIView.animate(withDuration: 1.5, delay: 0.0, options: [.autoreverse, .repeat], animations: {
            self.otherConstraint?.constant = -100
            self.constraint?.constant = -100
            self.view.layoutIfNeeded()
        })
    }
}
PlaygroundPage.current.liveView = ViewController()