diff --git a/qiskit/circuit/controlledgate.py b/qiskit/circuit/controlledgate.py
index c8b5b8a187be..8920e873ba53 100644
--- a/qiskit/circuit/controlledgate.py
+++ b/qiskit/circuit/controlledgate.py
@@ -243,11 +243,11 @@ def params(self, parameters):
         else:
             raise CircuitError("Controlled gate does not define base gate for extracting params")
 
-    def __deepcopy__(self, _memo=None):
+    def __deepcopy__(self, memo=None):
         cpy = copy.copy(self)
         cpy.base_gate = self.base_gate.copy()
         if self._definition:
-            cpy._definition = copy.deepcopy(self._definition, _memo)
+            cpy._definition = copy.deepcopy(self._definition, memo)
         return cpy
 
     @property
diff --git a/qiskit/circuit/library/standard_gates/equivalence_library.py b/qiskit/circuit/library/standard_gates/equivalence_library.py
index 5a519bf520e3..329c72b17cca 100644
--- a/qiskit/circuit/library/standard_gates/equivalence_library.py
+++ b/qiskit/circuit/library/standard_gates/equivalence_library.py
@@ -1210,6 +1210,7 @@
 lam = Parameter("lam")
 cu3_to_cu = QuantumCircuit(q)
 cu3_to_cu.cu(theta, phi, lam, 0, 0, 1)
+_sel.add_equivalence(CU3Gate(theta, phi, lam), cu3_to_cu)
 
 # XGate
 #
diff --git a/qiskit/circuit/library/standard_gates/u.py b/qiskit/circuit/library/standard_gates/u.py
index bb3e26495678..39c269def741 100644
--- a/qiskit/circuit/library/standard_gates/u.py
+++ b/qiskit/circuit/library/standard_gates/u.py
@@ -11,6 +11,7 @@
 # that they have been altered from the originals.
 
 """Two-pulse single-qubit gate."""
+import copy
 import math
 from cmath import exp
 from typing import Optional, Union
@@ -19,7 +20,6 @@
 from qiskit.circuit.gate import Gate
 from qiskit.circuit.parameterexpression import ParameterValueType
 from qiskit.circuit.quantumregister import QuantumRegister
-from qiskit.circuit.exceptions import CircuitError
 
 
 class UGate(Gate):
@@ -128,6 +128,35 @@ def __array__(self, dtype=complex):
         )
 
 
+class _CUGateParams(list):
+    # This awful class is to let `CUGate.params` have its keys settable (as
+    # `QuantumCircuit.assign_parameters` requires), while accounting for the problem that `CUGate`
+    # was defined to have a different number of parameters to its `base_gate`, which breaks
+    # `ControlledGate`'s assumptions, and would make most parametric `CUGate`s invalid.
+    #
+    # It's constructed only as part of the `CUGate.params` getter, and given that the general
+    # circuit model assumes that that's a directly mutable list that _must_ be kept in sync with the
+    # gate's requirements, we don't need this to support arbitrary mutation, just enough for
+    # `QuantumCircuit.assign_parameters` to work.
+
+    __slots__ = ("_gate",)
+
+    def __init__(self, gate):
+        super().__init__(gate._params)
+        self._gate = gate
+
+    def __setitem__(self, key, value):
+        if isinstance(key, slice):
+            raise TypeError("'CUGate' only supports setting parameters by single index")
+        super().__setitem__(key, value)
+        self._gate._params[key] = value
+        # Magic numbers: CUGate has 4 parameters, UGate has 3.
+        if key < 0:
+            key = 4 + key
+        if key < 3:
+            self._gate.base_gate.params[key] = value
+
+
 class CUGate(ControlledGate):
     r"""Controlled-U gate (4-parameter two-qubit gate).
 
@@ -273,33 +302,20 @@ def __array__(self, dtype=None):
 
     @property
     def params(self):
-        """Get parameters from base_gate.
-
-        Returns:
-            list: List of gate parameters.
-
-        Raises:
-            CircuitError: Controlled gate does not define a base gate
-        """
-        if self.base_gate:
-            # CU has one additional parameter to the U base gate
-            return self.base_gate.params + self._params
-        else:
-            raise CircuitError("Controlled gate does not define base gate for extracting params")
+        return _CUGateParams(self)
 
     @params.setter
     def params(self, parameters):
-        """Set base gate parameters.
-
-        Args:
-            parameters (list): The list of parameters to set.
-
-        Raises:
-            CircuitError: If controlled gate does not define a base gate.
-        """
-        # CU has one additional parameter to the U base gate
-        self._params = [parameters[-1]]
-        if self.base_gate:
-            self.base_gate.params = parameters[:-1]
-        else:
-            raise CircuitError("Controlled gate does not define base gate for extracting params")
+        # We need to skip `ControlledGate` in the inheritance tree, since it defines
+        # that all controlled gates are `(1-|c><c|).1 + |c><c|.base` for control-state `c`, which
+        # this class does _not_ satisfy (so it shouldn't really be a `ControlledGate`).
+        super(ControlledGate, type(self)).params.fset(self, parameters)
+        self.base_gate.params = parameters[:-1]
+
+    def __deepcopy__(self, memo=None):
+        # We have to override this because `ControlledGate` doesn't copy the `_params` list,
+        # assuming that `params` will be a view onto the base gate's `_params`.
+        memo = memo if memo is not None else {}
+        out = super().__deepcopy__(memo)
+        out._params = copy.deepcopy(out._params, memo)
+        return out
diff --git a/releasenotes/notes/fix-cu-assign-parameters-fbf6f21d3e0c8c0b.yaml b/releasenotes/notes/fix-cu-assign-parameters-fbf6f21d3e0c8c0b.yaml
new file mode 100644
index 000000000000..26cb3f6f6785
--- /dev/null
+++ b/releasenotes/notes/fix-cu-assign-parameters-fbf6f21d3e0c8c0b.yaml
@@ -0,0 +1,8 @@
+---
+fixes:
+  - |
+    :class:`.CUGate` will now behave correctly during calls to :meth:`.QuantumCircuit.assign_parameters`.
+    Previously, it would cause various odd errors, often some time after the initial circuit assignment.
+    See `#7326 <https://github.com/Qiskit/qiskit/issues/7326>`__, `#7410 <https://github.com/Qiskit/qiskit/issues/7410>`__,
+    `#9627 <https://github.com/Qiskit/qiskit/issues/9627>`__, `#10002 <https://github.com/Qiskit/qiskit/issues/10002>`__,
+    and `#10131 <https://github.com/Qiskit/qiskit/issues/10131>`__.
diff --git a/test/python/circuit/test_controlled_gate.py b/test/python/circuit/test_controlled_gate.py
index f0c33df0a600..6ebb9f9d75b2 100644
--- a/test/python/circuit/test_controlled_gate.py
+++ b/test/python/circuit/test_controlled_gate.py
@@ -1176,6 +1176,38 @@ def test_assign_parameters(self):
         self.assertEqual(bound1.parameters, {subs1[ptest]})
         self.assertEqual(bound2.parameters, {subs2[ptest]})
 
+    def test_assign_cugate(self):
+        """Test assignment of CUGate, which breaks the `ControlledGate` requirements by not being
+        equivalent to a direct control of its base gate."""
+
+        parameters = [Parameter("t"), Parameter("p"), Parameter("l"), Parameter("g")]
+        values = [0.1, 0.2, 0.3, 0.4]
+
+        qc = QuantumCircuit(2)
+        qc.cu(*parameters, 0, 1)
+        assigned = qc.assign_parameters(dict(zip(parameters, values)), inplace=False)
+
+        expected = QuantumCircuit(2)
+        expected.cu(*values, 0, 1)
+
+        self.assertEqual(assigned.data[0].operation.base_gate, expected.data[0].operation.base_gate)
+        self.assertEqual(assigned, expected)
+
+    def test_assign_nested_controlled_cu(self):
+        """Test assignment of an arbitrary controlled parametrised gate that appears through the
+        `Gate.control()` method on an already-controlled gate."""
+        theta = Parameter("t")
+        qc_c = QuantumCircuit(2)
+        qc_c.crx(theta, 1, 0)
+        custom_gate = qc_c.to_gate().control()
+        qc = QuantumCircuit(3)
+        qc.append(custom_gate, [0, 1, 2])
+        assigned = qc.assign_parameters({theta: 0.5})
+
+        # We're testing here that everything's been propagated through to the base gates; the `reps`
+        # is just some high number to make sure we unwrap any controlled and custom gates.
+        self.assertEqual(set(assigned.decompose(reps=3).parameters), set())
+
     @data(-1, 0, 1.4, "1", 4, 10)
     def test_improper_num_ctrl_qubits(self, num_ctrl_qubits):
         """